Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

Apply the reciprocal identity to $\csc (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

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

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

Step 3.2.2

Cancel the common factor.

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

Step 3.2.3

Rewrite the expression.

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

Step 3.3

Multiply $\frac{1}{\sin (x)}$ by $\frac{{\sin (x)}^{2}}{\cos (x)}$ .

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

Step 3.4

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

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

Step 4

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

$\tan (x)$

Step 5

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

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