Precalculus Examples

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

Step 1

Start on the left side.

$2 \cot^{2} (x) - \csc^{2} (x)$

Step 2

Simplify the expression.

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Step 2.1

Simplify each term.

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Step 2.1.1

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

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

Step 2.1.2

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

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

Step 2.1.3

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

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

Step 2.1.4

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

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

Step 2.1.5

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

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

Step 2.1.6

One to any power is one.

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

Step 2.2

Combine the numerators over the common denominator.

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

Step 2.3

Apply the cosine double-angle identity.

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

Step 3

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

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