Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

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.

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

Step 3.1.2

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

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

Step 3.2

Combine the numerators over the common denominator.

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

Step 3.3

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

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

Step 4

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

$2 \tan (x)$

Step 5

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

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