Precalculus Examples

$\frac{\sin (2 x)}{2 \sin^{2} (x)} = \cot (x)$

Step 1

Start on the left side.

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

Step 2

Simplify the expression.

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Step 2.1

Apply the sine double-angle identity.

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

Step 2.2

Cancel the common factor of $2$ .

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Step 2.2.1

Cancel the common factor.

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

Step 2.2.2

Rewrite the expression.

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

Step 2.3

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

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Step 2.3.1

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

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

Step 2.3.2

Cancel the common factors.

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Step 2.3.2.1

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

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

Step 2.3.2.2

Cancel the common factor.

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

Step 2.3.2.3

Rewrite the expression.

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

Step 3

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

$\cot (x)$

Step 4

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

$\frac{\sin (2 x)}{2 \sin^{2} (x)} = \cot (x)$ is an identity