Trigonometry Examples

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

Step 1

Start on the left side.

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

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.

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

Step 2.2

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

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

Step 2.3

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

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

Step 3

Simplify.

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Step 3.1

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

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

Step 3.2

Combine $2$ and $\frac{1}{\cos (x)}$ .

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

Step 3.3

Reduce the expression $\frac{{\sin (x)}^{2} \cos (x)}{{\cos (x)}^{2}}$ by cancelling the common factors.

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Step 3.3.1

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

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

Step 3.3.2

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

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

Step 3.3.3

Cancel the common factor.

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

Step 3.3.4

Rewrite the expression.

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

Step 3.4

Multiply the numerator by the reciprocal of the denominator.

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

Step 3.5

Combine.

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

Step 3.6

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

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

Step 4

Rewrite $\frac{\sin^{2} (x)}{2}$ as $\frac{1}{2} \cdot \sin^{2} (x)$ .

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

Step 5

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

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