Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Add fractions.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

Write each expression with a common denominator of $\tan (x) (1 - \sec (x))$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1 - \sec (x)}{\tan (x)}$ by $\frac{1 - \sec (x)}{1 - \sec (x)}$ .

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

Step 2.3.2

Multiply $\frac{\tan (x)}{1 - \sec (x)}$ by $\frac{\tan (x)}{\tan (x)}$ .

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

Step 2.3.3

Reorder the factors of $(1 - \sec (x)) \tan (x)$ .

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

Step 2.4

Combine the numerators over the common denominator.

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

Step 3

Simplify each term.

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

Step 4

Simplify denominator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 4.3

Reorder factors in $\tan (x) + \tan (x) (- \sec (x))$ .

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

Step 5

Apply Pythagorean identity.

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Step 5.1

Move $\tan^{2} (x)$ .

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

Step 5.2

Rearrange terms.

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

Step 5.3

Apply pythagorean identity.

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

Step 6

Convert to sines and cosines.

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Step 6.1

Apply the reciprocal identity to $\sec (x)$ .

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

Step 6.2

Apply the reciprocal identity to $\sec (x)$ .

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

Step 6.3

Apply the reciprocal identity to $\sec (x)$ .

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

Step 6.4

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 6.5

Write $\tan (x)$ in sines and cosines using the quotient identity.

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

Step 6.6

Apply the reciprocal identity to $\sec (x)$ .

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

Step 6.7

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

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

Step 6.8

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

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

Step 7

Simplify.

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Step 7.1

Multiply the numerator and denominator of the fraction by ${\cos (x)}^{2}$ .

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Step 7.1.1

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

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

Step 7.1.2

Combine.

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

Step 7.2

Apply the distributive property.

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

Step 7.3

Simplify by cancelling.

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Step 7.3.1

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

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Step 7.3.1.1

Cancel the common factor.

Step 7.3.1.2

Rewrite the expression.

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

Step 7.3.2

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

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Step 7.3.2.1

Cancel the common factor.

Step 7.3.2.2

Rewrite the expression.

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

Step 7.3.3

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

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Step 7.3.3.1

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

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

Step 7.3.3.2

Cancel the common factor.

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

Step 7.3.3.3

Rewrite the expression.

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

Step 7.4

Simplify the numerator.

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Step 7.4.1

Add $1^{2}$ and $1^{2}$ .

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

Step 7.4.2

Factor $2$ out of $2 \cdot 1^{2} + {\cos (x)}^{2} (- 2 \frac{1}{\cos (x)})$ .

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Step 7.4.2.1

Factor $2$ out of $2 \cdot 1^{2}$ .

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

Step 7.4.2.2

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

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

Step 7.4.2.3

Factor $2$ out of $2 (1^{2}) + 2 ({\cos (x)}^{2} (- \frac{1}{\cos (x)}))$ .

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

Step 7.4.3

One to any power is one.

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

Step 7.4.4

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

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Step 7.4.4.1

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

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

Step 7.4.4.2

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

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

Step 7.4.4.3

Cancel the common factor.

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

Step 7.4.4.4

Rewrite the expression.

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

Step 7.4.5

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

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

Step 7.4.6

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

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

Step 7.5

Simplify the denominator.

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Step 7.5.1

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

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Step 7.5.1.1

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

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

Step 7.5.1.2

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

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

Step 7.5.1.3

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

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

Step 7.5.2

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

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Step 7.5.2.1

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

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

Step 7.5.2.2

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

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

Step 7.5.2.3

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

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

Step 7.5.2.4

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

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

Step 7.5.2.5

Add $1$ and $1$ .

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

Step 7.5.3

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

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Step 7.5.3.1

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

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

Step 7.5.3.2

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

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

Step 7.5.3.3

Cancel the common factor.

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

Step 7.5.3.4

Rewrite the expression.

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

Step 7.5.4

Move the negative in front of the fraction.

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

Step 7.5.5

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

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Step 7.5.5.1

Multiply by $1$ .

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

Step 7.5.5.2

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

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

Step 7.5.5.3

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

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

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

Step 7.5.6

Write $1$ as a fraction with a common denominator.

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

Step 7.5.7

Combine the numerators over the common denominator.

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

Step 7.5.8

Combine exponents.

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Step 7.5.8.1

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

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

Step 7.5.8.2

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

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

Step 7.5.9

Reduce the expression by cancelling the common factors.

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Step 7.5.9.1

Reduce the expression $\frac{(\cos (x) - 1) \cos (x) \sin (x)}{\cos (x)}$ by cancelling the common factors.

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Step 7.5.9.1.1

Cancel the common factor.

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

Step 7.5.9.1.2

Rewrite the expression.

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

Step 7.5.9.2

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

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

Step 7.6

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

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Step 7.6.1

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

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

Step 7.6.2

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

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

Step 7.6.3

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

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

Step 7.6.4

Reorder terms.

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

Step 7.6.5

Cancel the common factor.

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

Step 7.6.6

Rewrite the expression.

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

Step 7.7

Move the negative in front of the fraction.

$- \frac{2}{\sin (x)}$

Step 8

Write $- 1$ as a fraction with denominator $1$ .

$\frac{- 1}{1} \cdot \frac{2}{\sin (x)}$

Step 9

Combine.

$\frac{- 1 \cdot 2}{1 \sin (x)}$

Step 10

Multiply $- 1$ by $2$ .

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

Step 11

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

$\frac{- 2}{\sin (x)}$

Step 12

Move the negative in front of the fraction.

$- \frac{2}{\sin (x)}$

Step 13

Now consider the right side of the equation.

$- 2 \csc (x)$

Step 14

Apply the reciprocal identity to $\csc (x)$ .

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

Step 15

Simplify.

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Step 15.1

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

$\frac{- 2}{\sin (x)}$

Step 15.2

Move the negative in front of the fraction.

$- \frac{2}{\sin (x)}$

Step 16

Because the two sides have been shown to be equivalent, the equation is an identity.

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