Precalculus Examples

$\frac{\cot (x) - \tan (x)}{\cot (x) + \tan (x)} = \cos (2 x)$

Step 1

Start on the left side.

$\frac{\cot (x) - \tan (x)}{\cot (x) + \tan (x)}$

Step 2

Simplify the expression.

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Step 2.1

Simplify the numerator.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.2

Simplify the denominator.

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Step 2.2.1

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

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

Step 2.2.2

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

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Combine.

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

Step 2.4

Apply the distributive property.

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

Step 2.5

Simplify by cancelling.

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Step 2.5.1

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

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Step 2.5.1.1

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

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

Step 2.5.1.2

Cancel the common factor.

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

Step 2.5.1.3

Rewrite the expression.

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

Step 2.5.2

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

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

Step 2.5.3

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

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

Step 2.5.4

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

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

Step 2.5.5

Add $1$ and $1$ .

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

Step 2.5.6

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

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Step 2.5.6.1

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

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

Step 2.5.6.2

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

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

Step 2.5.6.3

Cancel the common factor.

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

Step 2.5.6.4

Rewrite the expression.

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

Step 2.5.7

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

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

Step 2.5.8

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

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

Step 2.5.9

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

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

Step 2.5.10

Add $1$ and $1$ .

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

Step 2.5.11

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

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Step 2.5.11.1

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

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

Step 2.5.11.2

Cancel the common factor.

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

Step 2.5.11.3

Rewrite the expression.

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

Step 2.5.12

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

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

Step 2.5.13

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

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

Step 2.5.14

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

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

Step 2.5.15

Add $1$ and $1$ .

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

Step 2.5.16

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

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Step 2.5.16.1

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

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

Step 2.5.16.2

Cancel the common factor.

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

Step 2.5.16.3

Rewrite the expression.

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

Step 2.5.17

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

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

Step 2.5.18

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

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

Step 2.5.19

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

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

Step 2.5.20

Add $1$ and $1$ .

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

Step 2.6

Rearrange terms.

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

Step 2.7

Apply pythagorean identity.

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

Step 2.8

Divide $\cos^{2} (x) - \sin^{2} (x)$ by $1$ .

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

Step 2.9

Apply the cosine double-angle identity.

$\cos (2 x)$

Step 3

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

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