Trigonometry Examples

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

Step 1

Simplify the left side.

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Step 1.1

Simplify each term.

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Step 1.1.1

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

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

Step 1.1.2

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

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

Step 1.1.3

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.4

Apply the distributive property.

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

Step 1.1.5

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

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

Step 1.1.6

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

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Step 1.1.6.1

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

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

Step 1.1.6.2

Cancel the common factor.

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

Step 1.1.6.3

Rewrite the expression.

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

Step 1.1.7

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

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

Step 1.1.8

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

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

Step 1.1.9

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

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

Step 1.1.10

Multiply the numerator by the reciprocal of the denominator.

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

Step 1.1.11

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

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

Step 2

Simplify the right side.

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Step 2.1

Simplify $- 2 \csc (x)$ .

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

Move the negative in front of the fraction.

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

Step 3

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

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

Step 4

Apply the distributive property.

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

Step 5

Simplify.

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Step 5.1

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

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Step 5.1.1

Cancel the common factor.

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

Step 5.1.2

Rewrite the expression.

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

Step 5.2

Rewrite using the commutative property of multiplication.

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

Step 5.3

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

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Step 5.3.1

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

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

Step 5.3.2

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

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

Step 5.3.3

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

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

Step 5.3.4

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

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

Step 5.3.5

Add $1$ and $1$ .

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

Step 6

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

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Step 6.1

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

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

Step 6.2

Cancel the common factor.

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

Step 6.3

Rewrite the expression.

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

Step 7

Rewrite using the commutative property of multiplication.

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

Step 8

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

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Step 8.1

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

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

Step 8.2

Cancel the common factor.

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

Step 8.3

Rewrite the expression.

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

Step 9

Multiply $- 1$ by $2$ .

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

Step 10

Add $2$ to both sides of the equation.

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

Step 11

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

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Step 11.1

Simplify each term.

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Step 11.1.1

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

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

Step 11.1.2

Separate fractions.

$\cos (x) - 1 + \frac{\sin (x)}{1 - \frac{1}{\cos (x)}} \cdot \frac{\sin (x)}{\cos (x)} + 2 = 0$

Step 11.1.3

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

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

Step 11.1.4

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

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

Step 11.1.5

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

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

Step 11.2

To write $\cos (x)$ as a fraction with a common denominator, multiply by $\frac{1 - \sec (x)}{1 - \sec (x)}$ .

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

Step 11.3

Combine the numerators over the common denominator.

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

Step 11.4

Simplify the numerator.

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Step 11.4.1

Apply the distributive property.

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

Step 11.4.2

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

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

Step 11.4.3

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

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Step 11.4.3.1

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

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

Step 11.4.3.2

Add parentheses.

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

Step 11.4.3.3

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

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

Step 11.4.3.4

Cancel the common factors.

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

Step 11.4.4

Multiply $- 1$ by $1$ .

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

Step 11.5

Add $- 1$ and $2$ .

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

Step 12

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

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

Step 13

Multiply the numerator by the reciprocal of the denominator.

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

Step 14

Simplify the numerator.

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Step 14.1

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

$\frac{\cos (x) - 1 + \sin (x) (\frac{\sin (x)}{\cos (x)})}{1 - \sec (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2

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

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Step 14.2.1

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

$\frac{\cos (x) - 1 + \frac{\sin (x) \sin (x)}{\cos (x)}}{1 - \sec (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2.2

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

$\frac{\cos (x) - 1 + \frac{\sin (x) \sin (x)}{\cos (x)}}{1 - \sec (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2.3

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

$\frac{\cos (x) - 1 + \frac{\sin (x) \sin (x)}{\cos (x)}}{1 - \sec (x)} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 14.2.4

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

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

Step 14.2.5

Add $1$ and $1$ .

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

Step 15

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

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

Step 16

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

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Step 16.1

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

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

Step 16.2

Combine.

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

Step 17

Apply the distributive property.

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

Step 18

Simplify by cancelling.

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Step 18.1

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

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Step 18.1.1

Cancel the common factor.

Step 18.1.2

Rewrite the expression.

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

Step 18.2

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

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Step 18.2.1

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

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

Step 18.2.2

Cancel the common factor.

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

Step 18.2.3

Rewrite the expression.

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

Step 19

Simplify the numerator.

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Step 19.1

Rearrange terms.

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

Step 19.2

Rearrange terms.

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

Step 19.3

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

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

Step 19.4

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

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

Step 19.5

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

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

Step 19.6

Add $1$ and $1$ .

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

Step 19.7

Apply pythagorean identity.

$\frac{\cos (x) \cdot - 1 + 1}{\cos (x) \cdot 1 - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19.8

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

$\frac{- 1 \cdot \cos (x) + 1}{\cos (x) \cdot 1 - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 19.9

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

$\frac{- \cos (x) + 1}{\cos (x) \cdot 1 - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20

Reduce the expression by cancelling the common factors.

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Step 20.1

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

$\frac{- \cos (x) + 1}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2

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

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Step 20.2.1

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

$\frac{- (\cos (x)) + 1}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2.2

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

$\frac{- (\cos (x)) - 1 \cdot - 1}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2.3

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

$\frac{- (\cos (x) - 1)}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2.4

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

$\frac{- 1 (\cos (x) - 1)}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2.5

Cancel the common factor.

$\frac{- 1 (\cos (x) - 1)}{\cos (x) - 1} \cdot \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 20.2.6

Divide $- 1$ by $1$ .

$- \frac{1}{\cos (x)} + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 21

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

$- \sec (x) + \frac{1}{\cos (x)} = \frac{0}{\cos (x)}$

Step 22

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

$- \sec (x) + \sec (x) = \frac{0}{\cos (x)}$

Step 23

Separate fractions.

$- \sec (x) + \sec (x) = \frac{0}{1} \cdot \frac{1}{\cos (x)}$

Step 24

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

$- \sec (x) + \sec (x) = \frac{0}{1} \cdot \sec (x)$

Step 25

Divide $0$ by $1$ .

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

Step 26

Multiply $0$ by $\sec (x)$ .

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

Step 27

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

$0 = 0$

Step 28

Since $0 = 0$ , the equation will always be true for any value of $x$ .

All real numbers

Step 29

The result can be shown in multiple forms.

All real numbers

Interval Notation:

$(- \infty, \infty)$