Precalculus Examples Analytic Trigonometry

Simplify

$\frac{\tan (x) \cdot \sin (x)}{\sec^{2} (x) - 1}$

Step 1

Apply pythagorean identity.

$\frac{\tan (x) \cdot \sin (x)}{\tan^{2} (x)}$

Step 2

Cancel the common factor of $\tan (x)$ and $\tan^{2} (x)$ .

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Step 2.1

Factor $\tan (x)$ out of $\tan (x) \cdot \sin (x)$ .

$\frac{\tan (x) (\sin (x))}{\tan^{2} (x)}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

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

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

Step 2.2.2

Cancel the common factor.

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

Step 2.2.3

Rewrite the expression.

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

Step 3

Rewrite $\tan (x)$ in terms of sines and cosines.

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

Step 4

Multiply by the reciprocal of the fraction to divide by $\frac{\sin (x)}{\cos (x)}$ .

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

Step 5

Write $\sin (x)$ as a fraction with denominator $1$ .

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

$\cos (x)$

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