Trigonometry Examples

$\frac{\tan (x) + \sec (x) - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Step 1

Simplify the left side.

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Step 1.1

Simplify $\frac{\tan (x) + \sec (x) - 1}{\tan (x) - \sec (x) + 1}$ .

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Step 1.1.1

Simplify the numerator.

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Step 1.1.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\frac{\sin (x)}{\cos (x)} + \sec (x) - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Step 1.1.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\tan (x) - \sec (x) + 1} = \tan (x) + \sec (x)$

Step 1.1.2

Simplify the denominator.

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Step 1.1.2.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \sec (x) + 1} = \tan (x) + \sec (x)$

Step 1.1.2.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1} = \tan (x) + \sec (x)$

Step 1.1.3

Multiply the numerator and denominator of the fraction by $\cos (x)$ .

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Step 1.1.3.1

Multiply $\frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1}$ by $\frac{\cos (x)}{\cos (x)}$ .

$\frac{\cos (x)}{\cos (x)} \cdot \frac{\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1}{\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1} = \tan (x) + \sec (x)$

Step 1.1.3.2

Combine.

$\frac{\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1)}{\cos (x) (\frac{\sin (x)}{\cos (x)} - \frac{1}{\cos (x)} + 1)} = \tan (x) + \sec (x)$

Step 1.1.4

Apply the distributive property.

$\frac{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5

Simplify by cancelling.

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Step 1.1.5.1

Cancel the common factor of $\cos (x)$ .

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Step 1.1.5.1.1

Cancel the common factor.

Step 1.1.5.1.2

Rewrite the expression.

$\frac{\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.2

Cancel the common factor of $\cos (x)$ .

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Step 1.1.5.2.1

Cancel the common factor.

$\frac{\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.2.2

Rewrite the expression.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.3

Cancel the common factor of $\cos (x)$ .

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Step 1.1.5.3.1

Cancel the common factor.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.3.2

Rewrite the expression.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.4

Cancel the common factor of $\cos (x)$ .

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Step 1.1.5.4.1

Move the leading negative in $- \frac{1}{\cos (x)}$ into the numerator.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) \frac{- 1}{\cos (x)} + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.4.2

Cancel the common factor.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) + \cos (x) \frac{- 1}{\cos (x)} + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.5.4.3

Rewrite the expression.

$\frac{\sin (x) + 1 + \cos (x) \cdot - 1}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.6

Simplify the numerator.

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Step 1.1.6.1

Move $- 1$ to the left of $\cos (x)$ .

$\frac{\sin (x) + 1 - 1 \cdot \cos (x)}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.6.2

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x) \cdot 1} = \tan (x) + \sec (x)$

Step 1.1.7

Multiply $\cos (x)$ by $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 2

Simplify the right side.

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Step 2.1

Simplify each term.

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Step 2.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 2.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 3

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 4

Combine $\cos (x)$ and $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 5

Apply the distributive property.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 6

Cancel the common factor of $\cos (x)$ .

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Step 6.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 6.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 7

Cancel the common factor of $\cos (x)$ .

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Step 7.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 7.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + 1$

Step 8

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 9

Separate fractions.

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 10

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 11

Divide $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ by $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 12

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ and $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 13

Reorder factors in $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 14

Simplify each term.

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Step 14.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Step 14.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15

Simplify the left side.

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Step 15.1

Simplify $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Step 15.1.1

Simplify the numerator.

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Step 15.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15.1.1.2

Combine $\cos (x)$ and $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15.1.1.3

Reduce the expression by cancelling the common factors.

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Step 15.1.1.3.1

Reduce the expression $\frac{\cos (x)}{\cos (x)}$ by cancelling the common factors.

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Step 15.1.1.3.1.1

Cancel the common factor.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15.1.1.3.1.2

Rewrite the expression.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15.1.1.3.2

Rewrite the expression.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 15.1.2

Multiply $\sin (x) + 1 - \cos (x)$ by $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 16

Simplify the right side.

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Step 16.1

Simplify each term.

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Step 16.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 16.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 17

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 18

Combine $\cos (x)$ and $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 19

Apply the distributive property.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 20

Cancel the common factor of $\cos (x)$ .

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Step 20.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 20.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 21

Cancel the common factor of $\cos (x)$ .

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Step 21.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 21.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + 1$

Step 22

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 23

Separate fractions.

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 24

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 25

Divide $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ by $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 26

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ and $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 27

Reorder factors in $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 28

Simplify each term.

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Step 28.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Step 28.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29

Simplify the left side.

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Step 29.1

Simplify $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Step 29.1.1

Simplify the numerator.

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Step 29.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29.1.1.2

Combine $\cos (x)$ and $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29.1.1.3

Reduce the expression by cancelling the common factors.

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Step 29.1.1.3.1

Reduce the expression $\frac{\cos (x)}{\cos (x)}$ by cancelling the common factors.

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Step 29.1.1.3.1.1

Cancel the common factor.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29.1.1.3.1.2

Rewrite the expression.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29.1.1.3.2

Rewrite the expression.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 29.1.2

Multiply $\sin (x) + 1 - \cos (x)$ by $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 30

Simplify the right side.

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Step 30.1

Simplify each term.

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Step 30.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 30.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 31

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 32

Combine $\cos (x)$ and $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 33

Apply the distributive property.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 34

Cancel the common factor of $\cos (x)$ .

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Step 34.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 34.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 35

Cancel the common factor of $\cos (x)$ .

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Step 35.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 35.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + 1$

Step 36

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 37

Separate fractions.

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 38

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 39

Divide $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ by $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 40

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ and $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 41

Reorder factors in $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 42

Simplify each term.

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Step 42.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Step 42.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43

Simplify the left side.

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Step 43.1

Simplify $\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

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Step 43.1.1

Simplify the numerator.

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Step 43.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\cos (x) \frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43.1.1.2

Combine $\cos (x)$ and $\frac{1}{\cos (x)}$ .

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43.1.1.3

Reduce the expression by cancelling the common factors.

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Step 43.1.1.3.1

Reduce the expression $\frac{\cos (x)}{\cos (x)}$ by cancelling the common factors.

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Step 43.1.1.3.1.1

Cancel the common factor.

$\frac{\frac{\cos (x)}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43.1.1.3.1.2

Rewrite the expression.

$\frac{\frac{1}{1} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43.1.1.3.2

Rewrite the expression.

$\frac{1 (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 43.1.2

Multiply $\sin (x) + 1 - \cos (x)$ by $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 44

Simplify the right side.

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Step 44.1

Simplify each term.

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Step 44.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 44.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 45

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 46

Combine $\cos (x)$ and $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 47

Apply the distributive property.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 48

Cancel the common factor of $\cos (x)$ .

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Step 48.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 48.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 49

Cancel the common factor of $\cos (x)$ .

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Step 49.1

Cancel the common factor.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 49.2

Rewrite the expression.

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sin (x) + 1$

Step 50

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 51

Separate fractions.

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \frac{1}{\cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 52

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{1} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 53

Divide $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ by $1$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \sec (x) = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 54

Combine $\frac{\cos (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ and $\sec (x)$ .

$\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 55

Reorder factors in $\frac{\cos (x) (\sin (x) + 1 - \cos (x)) \sec (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 56

Simplify each term.

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Step 56.1

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \frac{1}{\cos (x)}$

Step 56.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) + \sec (x)$

Step 57

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 58

Multiply the numerator by the reciprocal of the denominator.

$\frac{\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 59

Cancel the common factor of $\cos (x)$ .

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Step 59.1

Factor $\cos (x)$ out of $\cos (x) \sec (x) (\sin (x) + 1 - \cos (x))$ .

$\frac{\cos (x) (\sec (x) (\sin (x) + 1 - \cos (x)))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 59.2

Cancel the common factor.

$\frac{\cos (x) (\sec (x) (\sin (x) + 1 - \cos (x)))}{\sin (x) - 1 + \cos (x)} \cdot \frac{1}{\cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 59.3

Rewrite the expression.

$\frac{\sec (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 60

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\frac{1}{\cos (x)} \cdot (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 61

Factor $\frac{1}{\cos (x)}$ out of $\frac{\frac{1}{\cos (x)} (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)}$ .

$\frac{1}{\cos (x)} \cdot \frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 62

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\tan (x)}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63

Simplify each term.

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Step 63.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.2

Rewrite $\frac{\frac{\sin (x)}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.3

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.4

Simplify the denominator.

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Step 63.4.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.4.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \frac{\sec (x)}{\cos (x)}$

Step 63.4.4

Add $1$ and $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sec (x)}{\cos (x)}$

Step 63.5

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{\frac{1}{\cos (x)}}{\cos (x)}$

Step 63.6

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}$

Step 63.7

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos (x) \cos (x)}$

Step 63.8

Simplify the denominator.

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Step 63.8.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos (x) \cos (x)}$

Step 63.8.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos (x) \cos (x)}$

Step 63.8.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{{\cos (x)}^{1 + 1}}$

Step 63.8.4

Add $1$ and $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 64

Simplify each term.

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Step 64.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{1}{\cos^{2} (x)}$

Step 64.2

Separate fractions.

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)}$

Step 64.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{1}{\cos (x)} \cdot \tan (x) + \frac{1}{\cos^{2} (x)}$

Step 64.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \sec (x) \tan (x) + \frac{1}{\cos^{2} (x)}$

Step 64.5

Rewrite $1$ as $1^{2}$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)}$

Step 64.6

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2}$

Step 64.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \sec (x) \tan (x) + \sec^{2} (x)$

Step 65

Divide each term in the equation by $\cos (x)$ .

$\frac{\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}}{\cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 66

Separate fractions.

$\frac{\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}}{1} \cdot \frac{1}{\cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 67

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}}{1} \cdot \sec (x) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 68

Divide $\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}$ by $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} \cdot \sec (x) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 69

Combine $\frac{\sin (x) + 1 - \cos (x)}{\cos (x) (\sin (x) - 1 + \cos (x))}$ and $\sec (x)$ .

$\frac{(\sin (x) + 1 - \cos (x)) \sec (x)}{\cos (x) (\sin (x) - 1 + \cos (x))} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 70

Separate fractions.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot \frac{\sec (x)}{\cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 71

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 72

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73

Simplify.

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Step 73.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot (\sec (x) (\frac{1}{\cos (x)})) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73.2

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot (\sec (x) \sec (x)) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73.3

Raise $\sec (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot (\sec (x) \sec (x)) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73.4

Raise $\sec (x)$ to the power of $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot (\sec (x) \sec (x)) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73.5

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot {\sec (x)}^{1 + 1} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 73.6

Add $1$ and $1$ .

$\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)} \cdot \sec^{2} (x) = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 74

Combine $\frac{\sin (x) + 1 - \cos (x)}{\sin (x) - 1 + \cos (x)}$ and $\sec^{2} (x)$ .

$\frac{(\sin (x) + 1 - \cos (x)) \sec^{2} (x)}{\sin (x) - 1 + \cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 75

Reorder factors in $\frac{(\sin (x) + 1 - \cos (x)) \sec^{2} (x)}{\sin (x) - 1 + \cos (x)}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sec (x) \tan (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76

Simplify each term.

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Step 76.1

Separate fractions.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{1} \cdot \frac{\sec (x)}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{1} \cdot \frac{\frac{1}{\cos (x)}}{\cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.3

Rewrite $\frac{\frac{1}{\cos (x)}}{\cos (x)}$ as a product.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{1} \cdot (\frac{1}{\cos (x)} \cdot \frac{1}{\cos (x)}) + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.4

Multiply $\frac{1}{\cos (x)}$ by $\frac{1}{\cos (x)}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\tan (x)}{1} \cdot \frac{1}{\cos (x) \cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.5

Divide $\tan (x)$ by $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \tan (x) (\frac{1}{\cos (x) \cos (x)}) + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.6

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x) \cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.7

Simplify the denominator.

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Step 76.7.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x) \cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.7.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x) \cos (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.7.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{{\cos (x)}^{1 + 1}} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.7.4

Add $1$ and $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos^{2} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.8

Combine.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x) \cdot 1}{\cos (x) \cos^{2} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.9

Multiply $\cos (x)$ by $\cos^{2} (x)$ by adding the exponents.

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Step 76.9.1

Multiply $\cos (x)$ by $\cos^{2} (x)$ .

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Step 76.9.1.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x) \cdot 1}{\cos (x) \cos^{2} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.9.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x) \cdot 1}{{\cos (x)}^{1 + 2}} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.9.2

Add $1$ and $2$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x) \cdot 1}{\cos^{3} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.10

Multiply $\sin (x)$ by $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sec^{2} (x)}{\cos (x)}$

Step 76.11

Simplify the numerator.

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Step 76.11.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{{(\frac{1}{\cos (x)})}^{2}}{\cos (x)}$

Step 76.11.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\frac{1^{2}}{\cos^{2} (x)}}{\cos (x)}$

Step 76.11.3

One to any power is one.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{\frac{1}{\cos^{2} (x)}}{\cos (x)}$

Step 76.12

Multiply the numerator by the reciprocal of the denominator.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)} \cdot \frac{1}{\cos (x)}$

Step 76.13

Combine.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{2} (x) \cos (x)}$

Step 76.14

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 76.14.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 76.14.1.1

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{2} (x) \cos (x)}$

Step 76.14.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{{\cos (x)}^{2 + 1}}$

Step 76.14.2

Add $2$ and $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1 \cdot 1}{\cos^{3} (x)}$

Step 76.15

Multiply $1$ by $1$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}$

Step 77

Simplify each term.

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Step 77.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{\sin (x)}{\cos (x) \cos^{2} (x)} + \frac{1}{\cos^{3} (x)}$

Step 77.2

Separate fractions.

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}$

Step 77.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1}{\cos^{2} (x)} \cdot \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 77.4

Rewrite $1$ as $1^{2}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \frac{1^{2}}{\cos^{2} (x)} \cdot \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 77.5

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = {(\frac{1}{\cos (x)})}^{2} \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 77.6

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\frac{\sec^{2} (x) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 78

Replace the $\sec^{2} (x)$ with $1 + \tan^{2} (x)$ based on the $\tan^{2} (x) + 1 = \sec^{2} (x)$ identity.

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} = \sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}$

Step 79

Multiply both sides by $\sin (x) - 1 + \cos (x)$ .

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} (\sin (x) - 1 + \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80

Simplify.

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Step 80.1

Simplify the left side.

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Step 80.1.1

Simplify $\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} (\sin (x) - 1 + \cos (x))$ .

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Step 80.1.1.1

Cancel the common factor of $\sin (x) - 1 + \cos (x)$ .

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Step 80.1.1.1.1

Cancel the common factor.

$\frac{(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x))}{\sin (x) - 1 + \cos (x)} (\sin (x) - 1 + \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.1.2

Rewrite the expression.

$(1 + \tan^{2} (x)) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.2

Rearrange terms.

$(\tan^{2} (x) + 1) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.3

Apply pythagorean identity.

$\sec^{2} (x) (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.4

Rewrite $\sec (x)$ in terms of sines and cosines.

${(\frac{1}{\cos (x)})}^{2} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.5

Simplify the expression.

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Step 80.1.1.5.1

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{1^{2}}{\cos^{2} (x)} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.5.2

One to any power is one.

$\frac{1}{\cos^{2} (x)} (\sin (x) + 1 - \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.6

Apply the distributive property.

$\frac{1}{\cos^{2} (x)} \sin (x) + \frac{1}{\cos^{2} (x)} \cdot 1 + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.7

Simplify.

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Step 80.1.1.7.1

Combine $\frac{1}{\cos^{2} (x)}$ and $\sin (x)$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} \cdot 1 + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.7.2

Multiply $\frac{1}{\cos^{2} (x)}$ by $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} (- \cos (x)) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.7.3

Rewrite using the commutative property of multiplication.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos^{2} (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.8

Simplify each term.

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Step 80.1.1.8.1

Cancel the common factor of $\cos (x)$ .

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Step 80.1.1.8.1.1

Move the leading negative in $- \frac{1}{\cos^{2} (x)}$ into the numerator.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos^{2} (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.8.1.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x) \cos (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.8.1.3

Cancel the common factor.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x) \cos (x)} \cos (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.8.1.4

Rewrite the expression.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} + \frac{- 1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.8.2

Move the negative in front of the fraction.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9

Simplify each term.

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Step 80.1.1.9.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x)}{\cos (x) \cos (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.2

Separate fractions.

$\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.3

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\frac{1}{\cos (x)} \tan (x) + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.5

Rewrite $1$ as $1^{2}$ .

$\sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.6

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2} - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.7

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \frac{1}{\cos (x)} = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.1.1.9.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2

Simplify the right side.

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Step 80.2.1

Simplify $(\sec^{2} (x) \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$ .

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Step 80.2.1.1

Simplify each term.

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Step 80.2.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = ({(\frac{1}{\cos (x)})}^{2} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.2

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1^{2}}{\cos^{2} (x)} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.3

One to any power is one.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1}{\cos^{2} (x)} \tan (x) + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.4

Rewrite $\tan (x)$ in terms of sines and cosines.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1}{\cos^{2} (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.5

Combine.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{2} (x) \cos (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.6

Multiply $\cos^{2} (x)$ by $\cos (x)$ by adding the exponents.

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Step 80.2.1.1.6.1

Multiply $\cos^{2} (x)$ by $\cos (x)$ .

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Step 80.2.1.1.6.1.1

Raise $\cos (x)$ to the power of $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{2} (x) \cos^{1} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.6.1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{{\cos (x)}^{2 + 1}} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.6.2

Add $2$ and $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{1 \sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.1.7

Multiply $\sin (x)$ by $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = (\frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$

Step 80.2.1.2

Expand $(\frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)}) (\sin (x) - 1 + \cos (x))$ by multiplying each term in the first expression by each term in the second expression.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin (x)}{\cos^{3} (x)} \sin (x) + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3

Simplify terms.

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Step 80.2.1.3.1

Simplify each term.

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Step 80.2.1.3.1.1

Multiply $\frac{\sin (x)}{\cos^{3} (x)} \sin (x)$ .

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Step 80.2.1.3.1.1.1

Combine $\frac{\sin (x)}{\cos^{3} (x)}$ and $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin (x) \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.1.2

Raise $\sin (x)$ to the power of $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{1} (x) \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.1.3

Raise $\sin (x)$ to the power of $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{1} (x) \sin^{1} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.1.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{{\sin (x)}^{1 + 1}}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.1.5

Add $1$ and $1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cdot - 1 + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.2

Combine $\frac{\sin (x)}{\cos^{3} (x)}$ and $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x) \cdot - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.3

Move $- 1$ to the left of $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{- 1 \cdot \sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.4

Move the negative in front of the fraction.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{3} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.5

Cancel the common factor of $\cos (x)$ .

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Step 80.2.1.3.1.5.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos (x) \cos^{2} (x)} \cos (x) + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.5.2

Cancel the common factor.

Step 80.2.1.3.1.5.3

Rewrite the expression.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{3} (x)} \sin (x) + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.6

Combine $\frac{1}{\cos^{3} (x)}$ and $\sin (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cdot - 1 + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.7

Combine $\frac{1}{\cos^{3} (x)}$ and $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} + \frac{- 1}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.8

Move the negative in front of the fraction.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{3} (x)} \cos (x)$

Step 80.2.1.3.1.9

Cancel the common factor of $\cos (x)$ .

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Step 80.2.1.3.1.9.1

Factor $\cos (x)$ out of $\cos^{3} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos (x) \cos^{2} (x)} \cos (x)$

Step 80.2.1.3.1.9.2

Cancel the common factor.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos (x) \cos^{2} (x)} \cos (x)$

Step 80.2.1.3.1.9.3

Rewrite the expression.

Step 80.2.1.3.2

Simplify terms.

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Step 80.2.1.3.2.1

Combine the opposite terms in $\frac{\sin^{2} (x)}{\cos^{3} (x)} - \frac{\sin (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{\sin (x)}{\cos^{3} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$ .

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Step 80.2.1.3.2.1.1

Add $- \frac{\sin (x)}{\cos^{3} (x)}$ and $\frac{\sin (x)}{\cos^{3} (x)}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + 0 - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.1.2

Add $\frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)}$ and $0$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} - \frac{1}{\cos^{3} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.2

Combine the numerators over the common denominator.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\sin^{2} (x) - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.3

Reorder $\sin^{2} (x)$ and $- 1$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 + \sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.4

Rewrite $- 1$ as $- 1 (1)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1) + \sin^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.5

Factor $- 1$ out of $\sin^{2} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1) - 1 (- \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.6

Factor $- 1$ out of $- 1 (1) - 1 (- \sin^{2} (x))$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1 (1 - \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.3.2.7

Rewrite $- 1 (1 - \sin^{2} (x))$ as $- (1 - \sin^{2} (x))$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- (1 - \sin^{2} (x))}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.4

Apply pythagorean identity.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- \cos^{2} (x)}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5

Simplify terms.

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Step 80.2.1.5.1

Simplify each term.

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Step 80.2.1.5.1.1

Cancel the common factor of $\cos^{2} (x)$ and $\cos^{3} (x)$ .

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Step 80.2.1.5.1.1.1

Factor $\cos^{2} (x)$ out of $- \cos^{2} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\cos^{2} (x) \cdot - 1}{\cos^{3} (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.1.1.2

Cancel the common factors.

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Step 80.2.1.5.1.1.2.1

Factor $\cos^{2} (x)$ out of $\cos^{3} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\cos^{2} (x) \cdot - 1}{\cos^{2} (x) \cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.1.1.2.2

Cancel the common factor.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{\cos^{2} (x) \cdot - 1}{\cos^{2} (x) \cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.1.1.2.3

Rewrite the expression.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = \frac{- 1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.1.2

Move the negative in front of the fraction.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2

Simplify each term.

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Step 80.2.1.5.2.1

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2.2

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \frac{\sin (x)}{\cos (x) \cos (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2.3

Separate fractions.

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2.4

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \frac{1}{\cos (x)} \tan (x) + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2.5

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \frac{1}{\cos^{2} (x)}$

Step 80.2.1.5.2.6

Rewrite $1$ as $1^{2}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \frac{1^{2}}{\cos^{2} (x)}$

Step 80.2.1.5.2.7

Rewrite $\frac{1^{2}}{\cos^{2} (x)}$ as ${(\frac{1}{\cos (x)})}^{2}$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + {(\frac{1}{\cos (x)})}^{2}$

Step 80.2.1.5.2.8

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\sec (x) \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81

Solve for $x$ .

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Step 81.1

Simplify the left side.

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Step 81.1.1

Simplify each term.

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Step 81.1.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} \tan (x) + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.2

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.3

Multiply $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 81.1.1.3.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin (x)}{\cos (x) \cos (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.3.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos^{1} (x) \cos (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.3.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.3.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.3.5

Add $1$ and $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \sec^{2} (x) - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.4

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + {(\frac{1}{\cos (x)})}^{2} - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.5

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1^{2}}{\cos^{2} (x)} - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.6

One to any power is one.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \sec (x) = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.1.1.7

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \sec (x) + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.2

Simplify the right side.

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Step 81.2.1

Simplify each term.

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Step 81.2.1.1

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \sec (x) \tan (x) + \sec^{2} (x)$

Step 81.2.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{1}{\cos (x)} \tan (x) + \sec^{2} (x)$

Step 81.2.1.3

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)} + \sec^{2} (x)$

Step 81.2.1.4

Multiply $\frac{1}{\cos (x)} \cdot \frac{\sin (x)}{\cos (x)}$ .

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Step 81.2.1.4.1

Multiply $\frac{1}{\cos (x)}$ by $\frac{\sin (x)}{\cos (x)}$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x) \cos (x)} + \sec^{2} (x)$

Step 81.2.1.4.2

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{1} (x) \cos (x)} + \sec^{2} (x)$

Step 81.2.1.4.3

Raise $\cos (x)$ to the power of $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{1} (x) \cos^{1} (x)} + \sec^{2} (x)$

Step 81.2.1.4.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{{\cos (x)}^{1 + 1}} + \sec^{2} (x)$

Step 81.2.1.4.5

Add $1$ and $1$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \sec^{2} (x)$

Step 81.2.1.5

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + {(\frac{1}{\cos (x)})}^{2}$

Step 81.2.1.6

Apply the product rule to $\frac{1}{\cos (x)}$ .

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1^{2}}{\cos^{2} (x)}$

Step 81.2.1.7

One to any power is one.

$\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)} = - \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)}$

Step 81.3

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)} - \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.4

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos^{2} (x)} + \cos (x) \frac{1}{\cos^{2} (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5

Simplify.

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Step 81.5.1

Cancel the common factor of $\cos (x)$ .

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Step 81.5.1.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\cos (x) \frac{\sin (x)}{\cos (x) \cos (x)} + \cos (x) \frac{1}{\cos^{2} (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5.1.2

Cancel the common factor.

Step 81.5.1.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos^{2} (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5.2

Cancel the common factor of $\cos (x)$ .

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Step 81.5.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5.2.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5.2.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \cos (x) (- \frac{1}{\cos (x)}) = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.5.3

Rewrite using the commutative property of multiplication.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - \cos (x) \frac{1}{\cos (x)} = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.6

Cancel the common factor of $\cos (x)$ .

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Step 81.6.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \cos (x) \cdot - 1 \frac{1}{\cos (x)} = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.6.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \cos (x) \cdot - 1 \frac{1}{\cos (x)} = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.6.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = \cos (x) (- \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos^{2} (x)} + \frac{1}{\cos^{2} (x)})$

Step 81.7

Apply the distributive property.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = \cos (x) (- \frac{1}{\cos (x)}) + \cos (x) \frac{\sin (x)}{\cos^{2} (x)} + \cos (x) \frac{1}{\cos^{2} (x)}$

Step 81.8

Simplify.

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Step 81.8.1

Rewrite using the commutative property of multiplication.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \cos (x) \frac{\sin (x)}{\cos^{2} (x)} + \cos (x) \frac{1}{\cos^{2} (x)}$

Step 81.8.2

Cancel the common factor of $\cos (x)$ .

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Step 81.8.2.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \cos (x) \frac{\sin (x)}{\cos (x) \cos (x)} + \cos (x) \frac{1}{\cos^{2} (x)}$

Step 81.8.2.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \cos (x) \frac{\sin (x)}{\cos (x) \cos (x)} + \cos (x) \frac{1}{\cos^{2} (x)}$

Step 81.8.2.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos^{2} (x)}$

Step 81.8.3

Cancel the common factor of $\cos (x)$ .

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Step 81.8.3.1

Factor $\cos (x)$ out of $\cos^{2} (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \cos (x)}$

Step 81.8.3.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x) \cos (x)}$

Step 81.8.3.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - \cos (x) \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.9

Cancel the common factor of $\cos (x)$ .

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Step 81.9.1

Factor $\cos (x)$ out of $- \cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = \cos (x) \cdot - 1 \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.9.2

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = \cos (x) \cdot - 1 \frac{1}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.9.3

Rewrite the expression.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.10

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.11

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.12

Simplify.

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Step 81.12.1

Cancel the common factor of $\cos (x)$ .

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Step 81.12.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.12.1.2

Rewrite the expression.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.12.2

Cancel the common factor of $\cos (x)$ .

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Step 81.12.2.1

Cancel the common factor.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.12.2.2

Rewrite the expression.

$\sin (x) + 1 + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.12.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - 1 \cdot \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.13

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.14

Apply the distributive property.

$\sin (x) + 1 - \cos (x) = \cos (x) \cdot - 1 + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.15

Simplify.

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Step 81.15.1

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.15.2

Cancel the common factor of $\cos (x)$ .

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Step 81.15.2.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.15.2.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.15.3

Cancel the common factor of $\cos (x)$ .

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Step 81.15.3.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.15.3.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Step 81.16

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Step 81.17

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.18

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.19

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.20

Cancel the common factor of $\cos (x)$ .

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Step 81.20.1

Cancel the common factor.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.20.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.21

Simplify each term.

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Step 81.21.1

Cancel the common factor of $\cos (x)$ .

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Step 81.21.1.1

Cancel the common factor.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.21.1.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.21.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Step 81.21.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.22

Simplify the left side.

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Step 81.22.1

Simplify each term.

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Step 81.22.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.22.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.23

Simplify the right side.

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Step 81.23.1

Simplify each term.

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Step 81.23.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 81.23.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.24

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.25

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.26

Simplify.

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Step 81.26.1

Cancel the common factor of $\cos (x)$ .

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Step 81.26.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.26.1.2

Rewrite the expression.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.26.2

Cancel the common factor of $\cos (x)$ .

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Step 81.26.2.1

Cancel the common factor.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.26.2.2

Rewrite the expression.

$\sin (x) + 1 + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.26.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - 1 \cdot \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.27

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.28

Apply the distributive property.

$\sin (x) + 1 - \cos (x) = \cos (x) \cdot - 1 + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.29

Simplify.

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Step 81.29.1

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.29.2

Cancel the common factor of $\cos (x)$ .

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Step 81.29.2.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.29.2.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.29.3

Cancel the common factor of $\cos (x)$ .

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Step 81.29.3.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.29.3.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Step 81.30

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Step 81.31

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.32

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.33

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.34

Cancel the common factor of $\cos (x)$ .

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Step 81.34.1

Cancel the common factor.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.34.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.35

Simplify each term.

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Step 81.35.1

Cancel the common factor of $\cos (x)$ .

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Step 81.35.1.1

Cancel the common factor.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.35.1.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.35.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Step 81.35.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.36

Simplify the left side.

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Step 81.36.1

Simplify each term.

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Step 81.36.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.36.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.37

Simplify the right side.

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Step 81.37.1

Simplify each term.

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Step 81.37.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 81.37.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.38

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.39

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.40

Simplify.

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Step 81.40.1

Cancel the common factor of $\cos (x)$ .

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Step 81.40.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.40.1.2

Rewrite the expression.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.40.2

Cancel the common factor of $\cos (x)$ .

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Step 81.40.2.1

Cancel the common factor.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.40.2.2

Rewrite the expression.

$\sin (x) + 1 + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.40.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - 1 \cdot \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.41

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.42

Apply the distributive property.

$\sin (x) + 1 - \cos (x) = \cos (x) \cdot - 1 + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.43

Simplify.

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Step 81.43.1

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.43.2

Cancel the common factor of $\cos (x)$ .

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Step 81.43.2.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.43.2.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.43.3

Cancel the common factor of $\cos (x)$ .

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Step 81.43.3.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.43.3.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Step 81.44

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Step 81.45

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.46

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.47

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.48

Cancel the common factor of $\cos (x)$ .

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Step 81.48.1

Cancel the common factor.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.48.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.49

Simplify each term.

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Step 81.49.1

Cancel the common factor of $\cos (x)$ .

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Step 81.49.1.1

Cancel the common factor.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.49.1.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.49.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Step 81.49.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.50

Simplify the left side.

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Step 81.50.1

Simplify each term.

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Step 81.50.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.50.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.51

Simplify the right side.

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Step 81.51.1

Simplify each term.

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Step 81.51.1.1

Rewrite $\tan (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \sec (x)$

Step 81.51.1.2

Rewrite $\sec (x)$ in terms of sines and cosines.

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.52

Multiply both sides of the equation by $\cos (x)$ .

$\cos (x) (\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} - 1) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.53

Apply the distributive property.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.54

Simplify.

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Step 81.54.1

Cancel the common factor of $\cos (x)$ .

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Step 81.54.1.1

Cancel the common factor.

$\cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.54.1.2

Rewrite the expression.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.54.2

Cancel the common factor of $\cos (x)$ .

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Step 81.54.2.1

Cancel the common factor.

$\sin (x) + \cos (x) \frac{1}{\cos (x)} + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.54.2.2

Rewrite the expression.

$\sin (x) + 1 + \cos (x) \cdot - 1 = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.54.3

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - 1 \cdot \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.55

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = \cos (x) (- 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)})$

Step 81.56

Apply the distributive property.

$\sin (x) + 1 - \cos (x) = \cos (x) \cdot - 1 + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.57

Simplify.

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Step 81.57.1

Move $- 1$ to the left of $\cos (x)$ .

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.57.2

Cancel the common factor of $\cos (x)$ .

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Step 81.57.2.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \cos (x) \frac{\sin (x)}{\cos (x)} + \cos (x) \frac{1}{\cos (x)}$

Step 81.57.2.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.57.3

Cancel the common factor of $\cos (x)$ .

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Step 81.57.3.1

Cancel the common factor.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + \cos (x) \frac{1}{\cos (x)}$

Step 81.57.3.2

Rewrite the expression.

$\sin (x) + 1 - \cos (x) = - 1 \cdot \cos (x) + \sin (x) + 1$

Step 81.58

Rewrite $- 1 \cos (x)$ as $- \cos (x)$ .

$\sin (x) + 1 - \cos (x) = - \cos (x) + \sin (x) + 1$

Step 81.59

Divide each term in the equation by $\cos (x)$ .

$\frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.60

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \frac{1}{\cos (x)} + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.61

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.62

Cancel the common factor of $\cos (x)$ .

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Step 81.62.1

Cancel the common factor.

$\tan (x) + \sec (x) + \frac{- \cos (x)}{\cos (x)} = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.62.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.63

Simplify each term.

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Step 81.63.1

Cancel the common factor of $\cos (x)$ .

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Step 81.63.1.1

Cancel the common factor.

$\tan (x) + \sec (x) - 1 = \frac{- \cos (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.63.1.2

Divide $- 1$ by $1$ .

$\tan (x) + \sec (x) - 1 = - 1 + \frac{\sin (x)}{\cos (x)} + \frac{1}{\cos (x)}$

Step 81.63.2

Convert from $\frac{\sin (x)}{\cos (x)}$ to $\tan (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \frac{1}{\cos (x)}$

Step 81.63.3

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$\tan (x) + \sec (x) - 1 = - 1 + \tan (x) + \sec (x)$

Step 81.64

Move all terms containing $x$ to the left side of the equation.

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Step 81.64.1

Subtract $\tan (x)$ from both sides of the equation.

$\tan (x) + \sec (x) - 1 - \tan (x) = - 1 + \sec (x)$

Step 81.64.2

Subtract $\sec (x)$ from both sides of the equation.

$\tan (x) + \sec (x) - 1 - \tan (x) - \sec (x) = - 1$

Step 81.64.3

Combine the opposite terms in $\tan (x) + \sec (x) - 1 - \tan (x) - \sec (x)$ .

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Step 81.64.3.1

Subtract $\tan (x)$ from $\tan (x)$ .

$0 + \sec (x) - 1 - \sec (x) = - 1$

Step 81.64.3.2

Add $0$ and $\sec (x)$ .

$\sec (x) - 1 - \sec (x) = - 1$

Step 81.64.3.3

Subtract $\sec (x)$ from $\sec (x)$ .

$0 - 1 = - 1$

Step 81.64.3.4

Subtract $1$ from $0$ .

$- 1 = - 1$

Step 81.65

Since $- 1 = - 1$ , the equation will always be true for any value of $x$ .

All real numbers

Step 82

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$