Precalculus Examples

$\frac{4 \sec (x)}{\csc (x)} + \frac{\sin (x)}{\cos (x)} = 5 \tan (x)$

Step 1

Start on the left side.

$\frac{4 \sec (x)}{\csc (x)} + \frac{\sin (x)}{\cos (x)}$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{4 \frac{1}{\cos (x)}}{\csc (x)} + \frac{\sin (x)}{\cos (x)}$

Step 2.2

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

$\frac{4 \frac{1}{\cos (x)}}{\frac{1}{\sin (x)}} + \frac{\sin (x)}{\cos (x)}$

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.

$4 \frac{1}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)}$

Step 3.1.2

Combine $4$ and $\frac{1}{\cos (x)}$ .

$\frac{4}{\cos (x)} \sin (x) + \frac{\sin (x)}{\cos (x)}$

Step 3.1.3

Combine $\frac{4}{\cos (x)}$ and $\sin (x)$ .

$\frac{4 \sin (x)}{\cos (x)} + \frac{\sin (x)}{\cos (x)}$

Step 3.2

Combine the numerators over the common denominator.

$\frac{4 \sin (x) + \sin (x)}{\cos (x)}$

Step 3.3

Add $4 \sin (x)$ and $\sin (x)$ .

$\frac{5 \sin (x)}{\cos (x)}$

Step 4

Rewrite $\frac{5 \sin (x)}{\cos (x)}$ as $5 \tan (x)$ .

$5 \tan (x)$

Step 5

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

$\frac{4 \sec (x)}{\csc (x)} + \frac{\sin (x)}{\cos (x)} = 5 \tan (x)$ is an identity