Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

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

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

Apply the product rule to $\frac{\sin (x)}{\cos (x)}$ .

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

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

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Step 4.2.1

Factor $\sin (x)$ out of $\frac{\sin (x)}{\cos (x)} \cdot \sin (x)$ .

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

Step 4.2.2

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

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

Step 4.2.3

Cancel the common factor.

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

Step 4.2.4

Rewrite the expression.

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

Step 4.3

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

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

Step 4.4

Multiply $\frac{\sin (x)}{\cos (x)}$ by $\frac{{\cos (x)}^{2}}{\sin (x)}$ .

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

Step 4.5

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

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Step 4.5.1

Cancel the common factor.

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

Step 4.5.2

Rewrite the expression.

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

Step 4.6

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

$\cos (x)$

Step 5

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

$\frac{\tan (x) \cdot \sin (x)}{\sec^{2} (x) - 1} = \cos (x)$ is an identity