Trigonometry Examples

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

Step 1

Start on the left side.

$\sin^{2} (x) \sec (x) \csc (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

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

Step 2.2

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

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

Step 3

Simplify.

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Step 3.1

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

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Step 3.1.1

Factor $\sin (x)$ out of ${\sin (x)}^{2} \frac{1}{\cos (x)}$ .

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

Step 3.1.2

Cancel the common factor.

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

Step 3.1.3

Rewrite the expression.

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

Step 3.2

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

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

Step 4

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

$\tan (x)$

Step 5

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

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