Trigonometry Examples

$\frac{\sec (x)}{\tan (x)} = \csc (x)$

Step 1

Start on the left side.

$\frac{\sec (x)}{\tan (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.2

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

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

Step 4

Rewrite $\frac{1}{\sin (x)}$ as $\csc (x)$ .

$\csc (x)$

Step 5

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

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