Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

One to any power is one.

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

Step 4.3

Simplify the denominator.

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Step 4.3.1

Apply the distributive property.

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

Step 4.3.2

Multiply $- 1$ by $- 1$ .

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

Step 4.3.3

Add $1$ and $1$ .

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

Step 4.3.4

To write $2$ as a fraction with a common denominator, multiply by $\frac{{\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Step 4.3.5

Combine the numerators over the common denominator.

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

Step 4.4

Combine.

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

Step 4.5

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

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

Step 4.6

Combine $\cos (x)$ and $\frac{- 1 + 2 {\cos (x)}^{2}}{{\cos (x)}^{2}}$ .

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

Step 4.7

Reduce the expression $\frac{\cos (x) (- 1 + 2 {\cos (x)}^{2})}{{\cos (x)}^{2}}$ by cancelling the common factors.

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Step 4.7.1

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

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

Step 4.7.2

Cancel the common factor.

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

Step 4.7.3

Rewrite the expression.

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

Step 4.8

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.9

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

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

Step 4.10

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

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

Step 4.11

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

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

Step 4.12

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

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

Step 4.13

Move the negative in front of the fraction.

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

Step 5

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

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

Step 6

Combine.

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

Step 7

Remove parentheses.

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

Step 8

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

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

Step 9

Move the negative in front of the fraction.

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

Step 10

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

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

Step 11

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

$\frac{\tan (x)}{1 - \tan^{2} (x)} = \frac{\sin (x) \cos (x)}{2 \cos^{2} (x) - 1}$ is an identity