Precalculus Examples

$\frac{\cot^{3} (x)}{\csc (x)} = \cos (x) (\csc^{2} (x) - 1)$

Step 1

Start on the right side.

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

Step 2

Apply pythagorean identity.

$\cos (x) \cot^{2} (x)$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 4

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

$\frac{\cos^{3} (x)}{\sin^{2} (x)}$

Step 5

Rewrite $\frac{\cos^{3} (x)}{\sin^{2} (x)}$ as $\frac{\cot^{3} (x)}{\csc (x)}$ .

$\frac{\cot^{3} (x)}{\csc (x)}$

Step 6

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

$\frac{\cot^{3} (x)}{\csc (x)} = \cos (x) (\csc^{2} (x) - 1)$ is an identity