Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify the denominator.

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Step 1.1.1

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

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

Step 1.1.2

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

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

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

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

Step 2.1.2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Apply the distributive property.

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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Step 7.1

Cancel the common factor.

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

Step 7.2

Rewrite the expression.

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

Step 8

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

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

Step 9

Separate fractions.

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

Step 10

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

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

Step 11

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

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

Step 12

Simplify the denominator.

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Step 12.1

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

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

Step 12.2

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

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

Step 13

Combine $\frac{\cos (x)}{\sec (x) - \tan (x)}$ and $\sec (x)$ .

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

Step 14

Simplify each term.

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Step 14.1

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

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

Step 14.2

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

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

Step 15

Simplify the left side.

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Step 15.1

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

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Step 15.1.1

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

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

Step 15.1.2

Simplify the denominator.

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Step 15.1.2.1

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

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

Step 15.1.2.2

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

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

Step 15.1.3

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

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

Step 15.1.4

Reduce the expression by cancelling the common factors.

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Step 15.1.4.1

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

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Step 15.1.4.1.1

Cancel the common factor.

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

Step 15.1.4.1.2

Rewrite the expression.

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

Step 15.1.4.2

Rewrite the expression.

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

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

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

Step 16.1.2

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

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

Step 17

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

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

Step 18

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

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

Step 19

Apply the distributive property.

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

Step 20

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

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Step 20.1

Cancel the common factor.

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

Step 20.2

Rewrite the expression.

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

Step 21

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

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Step 21.1

Cancel the common factor.

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

Step 21.2

Rewrite the expression.

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

Step 22

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

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

Step 23

Separate fractions.

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

Step 24

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

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

Step 25

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

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

Step 26

Simplify the denominator.

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Step 26.1

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

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

Step 26.2

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

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

Step 27

Combine $\frac{\cos (x)}{\sec (x) - \tan (x)}$ and $\sec (x)$ .

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

Step 28

Simplify each term.

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Step 28.1

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

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

Step 28.2

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

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

Step 29

Simplify the left side.

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Step 29.1

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

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Step 29.1.1

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

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

Step 29.1.2

Simplify the denominator.

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Step 29.1.2.1

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

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

Step 29.1.2.2

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

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

Step 29.1.3

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

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

Step 29.1.4

Reduce the expression by cancelling the common factors.

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Step 29.1.4.1

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

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Step 29.1.4.1.1

Cancel the common factor.

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

Step 29.1.4.1.2

Rewrite the expression.

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

Step 29.1.4.2

Rewrite the expression.

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

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

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

Step 30.1.2

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

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

Step 31

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

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

Step 32

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

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

Step 33

Apply the distributive property.

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

Step 34

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

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Step 34.1

Cancel the common factor.

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

Step 34.2

Rewrite the expression.

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

Step 35

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

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Step 35.1

Cancel the common factor.

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

Step 35.2

Rewrite the expression.

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

Step 36

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

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

Step 37

Separate fractions.

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

Step 38

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

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

Step 39

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

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

Step 40

Simplify the denominator.

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Step 40.1

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

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

Step 40.2

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

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

Step 41

Combine $\frac{\cos (x)}{\sec (x) - \tan (x)}$ and $\sec (x)$ .

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

Step 42

Simplify each term.

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Step 42.1

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

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

Step 42.2

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

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

Step 43

Simplify the left side.

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Step 43.1

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

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Step 43.1.1

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

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

Step 43.1.2

Simplify the denominator.

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Step 43.1.2.1

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

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

Step 43.1.2.2

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

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

Step 43.1.3

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

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

Step 43.1.4

Reduce the expression by cancelling the common factors.

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Step 43.1.4.1

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

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Step 43.1.4.1.1

Cancel the common factor.

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

Step 43.1.4.1.2

Rewrite the expression.

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

Step 43.1.4.2

Rewrite the expression.

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

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

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

Step 44.1.2

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

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

Step 45

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

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

Step 46

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

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

Step 47

Apply the distributive property.

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

Step 48

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

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Step 48.1

Cancel the common factor.

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

Step 48.2

Rewrite the expression.

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

Step 49

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

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Step 49.1

Cancel the common factor.

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

Step 49.2

Rewrite the expression.

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

Step 50

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

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

Step 51

Separate fractions.

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

Step 52

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

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

Step 53

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

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

Step 54

Simplify the denominator.

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Step 54.1

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

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

Step 54.2

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

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

Step 55

Combine $\frac{\cos (x)}{\sec (x) - \tan (x)}$ and $\sec (x)$ .

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

Step 56

Simplify each term.

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Step 56.1

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

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

Step 56.2

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

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

Step 57

Multiply both sides by $\sec (x) - \tan (x)$ .

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

Step 58

Simplify.

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Step 58.1

Simplify the left side.

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Step 58.1.1

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

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Step 58.1.1.1

Cancel the common factor of $\sec (x) - \tan (x)$ .

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Step 58.1.1.1.1

Cancel the common factor.

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

Step 58.1.1.1.2

Rewrite the expression.

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

Step 58.1.1.2

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 58.1.1.2.1

Reorder $\cos (x)$ and $\sec (x)$ .

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

Step 58.1.1.2.2

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

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

Step 58.1.1.2.3

Cancel the common factors.

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

Step 58.2

Simplify the right side.

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Step 58.2.1

Simplify $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ .

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Step 58.2.1.1

Expand $(\sec (x) + \tan (x)) (\sec (x) - \tan (x))$ using the FOIL Method.

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Step 58.2.1.1.1

Apply the distributive property.

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

Step 58.2.1.1.2

Apply the distributive property.

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

Step 58.2.1.1.3

Apply the distributive property.

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

Step 58.2.1.2

Simplify terms.

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Step 58.2.1.2.1

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

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Step 58.2.1.2.1.1

Reorder the factors in the terms $\sec (x) (- \tan (x))$ and $\tan (x) \sec (x)$ .

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

Step 58.2.1.2.1.2

Add $- \sec (x) \tan (x)$ and $\sec (x) \tan (x)$ .

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

Step 58.2.1.2.1.3

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

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

Step 58.2.1.2.2

Simplify each term.

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Step 58.2.1.2.2.1

Multiply $\sec (x) \sec (x)$ .

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Step 58.2.1.2.2.1.1

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

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

Step 58.2.1.2.2.1.2

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

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

Step 58.2.1.2.2.1.3

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

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

Step 58.2.1.2.2.1.4

Add $1$ and $1$ .

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

Step 58.2.1.2.2.2

Rewrite using the commutative property of multiplication.

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

Step 58.2.1.2.2.3

Multiply $- \tan (x) \tan (x)$ .

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Step 58.2.1.2.2.3.1

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

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

Step 58.2.1.2.2.3.2

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

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

Step 58.2.1.2.2.3.3

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

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

Step 58.2.1.2.2.3.4

Add $1$ and $1$ .

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

Step 58.2.1.3

Apply pythagorean identity.

$1 = 1$

Step 59

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

All real numbers

Step 60

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$