Trigonometry Examples

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

Step 1

Start on the left side.

$\frac{\tan (x) \cot (x)}{\csc (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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Step 3.2.1

Cancel the common factor.

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

Step 3.2.2

Rewrite the expression.

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

Step 3.3

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

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Step 3.3.1

Cancel the common factor.

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

Step 3.3.2

Rewrite the expression.

$1 \sin (x)$

Step 3.4

Multiply $\sin (x)$ by $1$ .

$\sin (x)$

Step 4

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

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