Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

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

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

Step 3

Combine.

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

Step 4

Simplify numerator.

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Step 4.1

Apply the distributive property.

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

Step 4.2

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

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

Step 5

Simplify denominator.

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Step 5.1

Expand $(1 - \sec (x)) (1 + \sec (x))$ using the FOIL Method.

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Step 5.1.1

Apply the distributive property.

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

Step 5.1.2

Apply the distributive property.

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

Step 5.1.3

Apply the distributive property.

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

Step 5.2

Simplify and combine like terms.

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

Step 6

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

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

Step 7

Combine.

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

Step 8

Apply the distributive property.

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

Step 9

Multiply $1 - {\sec (x)}^{2}$ by $1$ .

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

Step 10

Apply Pythagorean identity.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

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

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

Step 10.5

Apply pythagorean identity.

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

Step 11

Convert to sines and cosines.

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Step 11.1

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

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

Step 11.2

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

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

Step 11.3

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

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

Step 11.4

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

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

Step 11.5

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

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

Step 11.6

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

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

Step 11.7

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

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

Step 12

Simplify.

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Step 12.1

Reorder the factors of $- \frac{\sin (x)}{\cos (x)} \cdot \frac{1}{\cos (x)}$ .

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

Step 12.2

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

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

Step 12.3

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

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

Step 12.4

Combine the numerators over the common denominator.

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

Step 12.5

Simplify each term.

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Step 12.5.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 12.5.2

Apply the distributive property.

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

Step 12.5.3

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

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

Step 12.5.4

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

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Step 12.5.4.1

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

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

Step 12.5.4.2

Cancel the common factor.

Step 12.5.4.3

Rewrite the expression.

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

Step 12.5.5

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

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

Step 12.5.6

Multiply the numerator by the reciprocal of the denominator.

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

Step 12.5.7

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

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Step 12.5.7.1

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

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

Step 12.5.7.2

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

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

Step 12.5.7.3

Cancel the common factor.

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

Step 12.5.7.4

Rewrite the expression.

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

Step 12.5.8

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

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

Step 12.5.9

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

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

Step 12.5.10

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

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

Step 12.5.11

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

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

Step 12.5.12

Add $1$ and $1$ .

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

Step 12.5.13

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

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

Step 12.5.14

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

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Step 12.5.14.1

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

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

Step 12.5.14.2

Cancel the common factors.

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Step 12.5.14.2.1

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

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

Step 12.5.14.2.2

Cancel the common factor.

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

Step 12.5.14.2.3

Rewrite the expression.

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

Step 12.5.15

Move the negative in front of the fraction.

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

Step 12.5.16

Apply the distributive property.

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

Step 12.5.17

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

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Step 12.5.17.1

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

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

Step 12.5.17.2

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

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

Step 12.5.17.3

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

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

Step 12.5.17.4

Cancel the common factor.

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

Step 12.5.17.5

Rewrite the expression.

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

Step 12.5.18

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

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Step 12.5.18.1

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

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

Step 12.5.18.2

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

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

Step 12.5.18.3

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

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

Step 12.5.18.4

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

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

Step 12.5.18.5

Cancel the common factor.

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

Step 12.5.18.6

Rewrite the expression.

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

Step 12.5.19

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

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Step 12.5.19.1

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

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

Step 12.5.19.2

Cancel the common factor.

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

Step 12.5.19.3

Rewrite the expression.

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

Step 12.5.20

Simplify each term.

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Step 12.5.20.1

Move the negative in front of the fraction.

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

Step 12.5.20.2

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

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Step 12.5.20.2.1

Multiply $- 1$ by $- 1$ .

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

Step 12.5.20.2.2

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

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

Step 12.5.20.3

Move the negative in front of the fraction.

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

Step 12.5.20.4

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

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Step 12.5.20.4.1

Multiply $- 1$ by $- 1$ .

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

Step 12.5.20.4.2

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

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

Step 12.6

Combine the numerators over the common denominator.

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

Step 12.7

Add $\cos (x)$ and $\cos (x)$ .

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

Step 12.8

Add $- 1$ and $1$ .

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

Step 12.9

Add $2 \cos (x)$ and $0$ .

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

Step 13

Rewrite $\frac{2 \cos (x)}{\sin (x)}$ as $2 \cot (x)$ .

$2 \cot (x)$

Step 14

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

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