Precalculus Examples

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

Step 1

Start on the left side.

$\tan (x) \cos (x) + \csc (x) \sin^{2} (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$\frac{\sin (x)}{\cos (x)} \cos (x) + \csc (x) \sin^{2} (x)$

Step 2.2

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

$\frac{\sin (x)}{\cos (x)} \cos (x) + \frac{1}{\sin (x)} \sin^{2} (x)$

Step 3

Simplify.

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Step 3.1

Simplify each term.

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Step 3.1.1

Cancel the common factor of $\cos (x)$ .

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Step 3.1.1.1

Cancel the common factor.

$\frac{\sin (x)}{\cos (x)} \cos (x) + \frac{1}{\sin (x)} {\sin (x)}^{2}$

Step 3.1.1.2

Rewrite the expression.

$\sin (x) + \frac{1}{\sin (x)} {\sin (x)}^{2}$

Step 3.1.2

Cancel the common factor of $\sin (x)$ .

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Step 3.1.2.1

Factor $\sin (x)$ out of ${\sin (x)}^{2}$ .

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

Step 3.1.2.2

Cancel the common factor.

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

Step 3.1.2.3

Rewrite the expression.

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

Step 3.2

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

$2 \sin (x)$

Step 4

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

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