Trigonometry Examples

$13 \sec (θ) \sin (θ) = 13 \cot (θ) \tan^{2} (θ)$

Step 1

Start on the right side.

$13 \cot (θ) \tan^{2} (θ)$

Step 2

Convert to sines and cosines.

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Step 2.1

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

$13 \frac{\cos (θ)}{\sin (θ)} \tan^{2} (θ)$

Step 2.2

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

$13 \frac{\cos (θ)}{\sin (θ)} {(\frac{\sin (θ)}{\cos (θ)})}^{2}$

Step 2.3

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

$13 \frac{\cos (θ)}{\sin (θ)} \frac{\sin^{2} (θ)}{\cos^{2} (θ)}$

Step 3

Simplify.

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Step 3.1

Combine $13$ and $\frac{\cos (θ)}{\sin (θ)}$ .

$\frac{13 \cos (θ)}{\sin (θ)} \cdot \frac{{\sin (θ)}^{2}}{{\cos (θ)}^{2}}$

Step 3.2

Combine.

$\frac{13 \cos (θ) {\sin (θ)}^{2}}{\sin (θ) {\cos (θ)}^{2}}$

Step 3.3

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

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Step 3.3.1

Factor $\cos (θ)$ out of $13 \cos (θ) {\sin (θ)}^{2}$ .

$\frac{\cos (θ) (13 {\sin (θ)}^{2})}{\sin (θ) {\cos (θ)}^{2}}$

Step 3.3.2

Cancel the common factors.

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Step 3.3.2.1

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

$\frac{\cos (θ) (13 {\sin (θ)}^{2})}{\cos (θ) (\sin (θ) \cos (θ))}$

Step 3.3.2.2

Cancel the common factor.

$\frac{\cos (θ) (13 {\sin (θ)}^{2})}{\cos (θ) (\sin (θ) \cos (θ))}$

Step 3.3.2.3

Rewrite the expression.

$\frac{13 {\sin (θ)}^{2}}{\sin (θ) \cos (θ)}$

Step 3.4

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

$\frac{13 \sin (θ)}{\cos (θ)}$

Step 4

Rewrite $\frac{13 \sin (θ)}{\cos (θ)}$ as $13 \sec (θ) \sin (θ)$ .

$13 \sec (θ) \sin (θ)$

Step 5

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

$13 \sec (θ) \sin (θ) = 13 \cot (θ) \tan^{2} (θ)$ is an identity