Trigonometry Examples

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

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

Step 1

Start on the left side.

\frac{\tan (x)}{\sec (x)} + \frac{\cot (x)}{\csc (x)}

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

Step 2

Convert to sines and cosines.

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Step 2.1

Write

\tan (x)

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

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

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

Step 2.2

Apply the reciprocal identity to

\sec (x)

$\sec (x)$ .

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

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

Step 2.3

Write

\cot (x)

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

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

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

Step 2.4

Apply the reciprocal identity to

\csc (x)

$\csc (x)$ .

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

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

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

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

Step 3

Simplify each term.

\sin (x) + \cos (x)

$\sin (x) + \cos (x)$

Step 4

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

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

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