Calculus Examples

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

Step 1

Start on the left side.

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

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

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

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

Step 3

Simplify each term.

$\sin (θ) + \cos (θ)$

Step 4

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

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