Precalculus Examples

$\frac{\csc (u) \sin (u)}{\cot (u)} = \tan (u)$

Step 1

Start on the left side.

$\frac{\csc (u) \sin (u)}{\cot (u)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{\frac{1}{\sin (u)} \sin (u)}{\cot (u)}$

Step 2.2

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

$\frac{\frac{1}{\sin (u)} \sin (u)}{\frac{\cos (u)}{\sin (u)}}$

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

$\frac{1}{\sin (u)} \sin (u) \frac{\sin (u)}{\cos (u)}$

Step 3.2

Cancel the common factor of $\sin (u)$ .

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Step 3.2.1

Cancel the common factor.

$\frac{1}{\sin (u)} \sin (u) \frac{\sin (u)}{\cos (u)}$

Step 3.2.2

Rewrite the expression.

$1 \frac{\sin (u)}{\cos (u)}$

Step 3.3

Multiply $\frac{\sin (u)}{\cos (u)}$ by $1$ .

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

Step 4

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

$\tan (u)$

Step 5

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

$\frac{\csc (u) \sin (u)}{\cot (u)} = \tan (u)$ is an identity