Trigonometry Examples

$\frac{\csc (x)}{\sec (x)} + \frac{\cos (x)}{\sin (x)} = 2 \cot (x)$

Step 1

Start on the left side.

$\frac{\csc (x)}{\sec (x)} + \frac{\cos (x)}{\sin (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.1.2

Combine $\frac{1}{\sin (x)}$ and $\cos (x)$ .

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

Add $\cos (x)$ and $\cos (x)$ .

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

Step 4

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

$2 \cot (x)$

Step 5

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

$\frac{\csc (x)}{\sec (x)} + \frac{\cos (x)}{\sin (x)} = 2 \cot (x)$ is an identity