Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Rearrange terms.

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

Step 2.2

Apply pythagorean identity.

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

Step 2.3

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

$\frac{2 \tan (x)}{\sec (x) \sec (x)}$

Step 2.4

Separate fractions.

$\frac{2}{\sec (x)} \cdot \frac{\tan (x)}{\sec (x)}$

Step 2.5

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

$\frac{2}{\sec (x)} \cdot \frac{\tan (x)}{\frac{1}{\cos (x)}}$

Step 2.6

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

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

Step 2.7

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 2.8

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 2.9

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

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Step 2.9.1

Cancel the common factor.

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

Step 2.9.2

Rewrite the expression.

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

Step 2.10

Separate fractions.

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

Step 2.11

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

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

Step 2.12

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\cos (x)}$ .

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

Step 2.13

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

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

Step 2.14

Reorder $\frac{2}{1} \cos (x)$ and $\sin (x)$ .

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

Step 2.15

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

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

Step 2.16

Divide $2$ by $1$ .

$2 \sin (x) \cos (x)$

Step 2.17

Apply the sine double-angle identity.

$\sin (2 x)$

Step 3

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

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