Precalculus Examples

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

Step 1

Rearrange terms.

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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

Step 4

Simplify the denominator.

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Step 4.1

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

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

Step 4.2

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

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

Step 4.3

One to any power is one.

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

Step 5

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

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

Step 6

Multiply the numerator by the reciprocal of the denominator.

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

Step 7

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

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Step 7.1

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

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

Step 7.2

Cancel the common factor.

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

Step 7.3

Rewrite the expression.

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

Step 8

Apply the sine double-angle identity.

$\sin (2 x) \cdot \sin (2 x)$

Step 9

Multiply $\sin (2 x) \sin (2 x)$ .

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Step 9.1

Raise $\sin (2 x)$ to the power of $1$ .

$\sin^{1} (2 x) \sin (2 x)$

Step 9.2

Raise $\sin (2 x)$ to the power of $1$ .

$\sin^{1} (2 x) \sin^{1} (2 x)$

Step 9.3

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

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

Step 9.4

Add $1$ and $1$ .

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