Trigonometry Examples

$8 \sec (x) \sin (x) = 8 \cot (x) \tan^{2} (x)$

Step 1

Start on the right side.

$8 \cot (x) \tan^{2} (x)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$8 \frac{\cos (x)}{\sin (x)} \tan^{2} (x)$

Step 2.2

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

$8 \frac{\cos (x)}{\sin (x)} {(\frac{\sin (x)}{\cos (x)})}^{2}$

Step 2.3

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

$8 \frac{\cos (x)}{\sin (x)} \frac{\sin^{2} (x)}{\cos^{2} (x)}$

Step 3

Simplify.

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Step 3.1

Combine $8$ and $\frac{\cos (x)}{\sin (x)}$ .

$\frac{8 \cos (x)}{\sin (x)} \cdot \frac{{\sin (x)}^{2}}{{\cos (x)}^{2}}$

Step 3.2

Combine.

$\frac{8 \cos (x) {\sin (x)}^{2}}{\sin (x) {\cos (x)}^{2}}$

Step 3.3

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

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Step 3.3.1

Factor $\cos (x)$ out of $8 \cos (x) {\sin (x)}^{2}$ .

$\frac{\cos (x) (8 {\sin (x)}^{2})}{\sin (x) {\cos (x)}^{2}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

$\frac{\cos (x) (8 {\sin (x)}^{2})}{\cos (x) (\sin (x) \cos (x))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{\cos (x) (8 {\sin (x)}^{2})}{\cos (x) (\sin (x) \cos (x))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{8 {\sin (x)}^{2}}{\sin (x) \cos (x)}$

Step 3.4

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

$\frac{8 \sin (x)}{\cos (x)}$

Step 4

Rewrite $\frac{8 \sin (x)}{\cos (x)}$ as $8 \sec (x) \sin (x)$ .

$8 \sec (x) \sin (x)$

Step 5

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

$8 \sec (x) \sin (x) = 8 \cot (x) \tan^{2} (x)$ is an identity