Trigonometry Examples

$9 \frac{\cot^{2} (θ)}{\csc (θ)} \sec^{2} (θ) = 9 \tan (θ) \cos (θ) \csc^{2} (θ)$

Step 1

Start on the left side.

$9 \frac{\cot^{2} (θ)}{\csc (θ)} \sec^{2} (θ)$

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.

$9 \frac{{(\frac{\cos (θ)}{\sin (θ)})}^{2}}{\csc (θ)} \sec^{2} (θ)$

Step 2.2

Apply the reciprocal identity to $\csc (θ)$ .

$9 \frac{{(\frac{\cos (θ)}{\sin (θ)})}^{2}}{\frac{1}{\sin (θ)}} \sec^{2} (θ)$

Step 2.3

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

$9 \frac{{(\frac{\cos (θ)}{\sin (θ)})}^{2}}{\frac{1}{\sin (θ)}} {(\frac{1}{\cos (θ)})}^{2}$

Step 2.4

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

$9 \frac{\frac{\cos^{2} (θ)}{\sin^{2} (θ)}}{\frac{1}{\sin (θ)}} {(\frac{1}{\cos (θ)})}^{2}$

Step 2.5

Apply the product rule to $\frac{1}{\cos (θ)}$ .

$9 \frac{\frac{\cos^{2} (θ)}{\sin^{2} (θ)}}{\frac{1}{\sin (θ)}} \frac{1^{2}}{\cos^{2} (θ)}$

Step 3

Simplify.

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Step 3.1

One to any power is one.

$9 \frac{\frac{{\cos (θ)}^{2}}{{\sin (θ)}^{2}}}{\frac{1}{\sin (θ)}} \frac{1}{{\cos (θ)}^{2}}$

Step 3.2

Multiply the numerator by the reciprocal of the denominator.

$9 (\frac{{\cos (θ)}^{2}}{{\sin (θ)}^{2}} \sin (θ)) \frac{1}{{\cos (θ)}^{2}}$

Step 3.3

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

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Step 3.3.1

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

$9 (\frac{{\cos (θ)}^{2}}{\sin (θ) \sin (θ)} \sin (θ)) \frac{1}{{\cos (θ)}^{2}}$

Step 3.3.2

Cancel the common factor.

$9 (\frac{{\cos (θ)}^{2}}{\sin (θ) \sin (θ)} \sin (θ)) \frac{1}{{\cos (θ)}^{2}}$

Step 3.3.3

Rewrite the expression.

$9 \frac{{\cos (θ)}^{2}}{\sin (θ)} \frac{1}{{\cos (θ)}^{2}}$

Step 3.4

Combine $9$ and $\frac{{\cos (θ)}^{2}}{\sin (θ)}$ .

$\frac{9 {\cos (θ)}^{2}}{\sin (θ)} \cdot \frac{1}{{\cos (θ)}^{2}}$

Step 3.5

Combine.

$\frac{9 {\cos (θ)}^{2} \cdot 1}{\sin (θ) {\cos (θ)}^{2}}$

Step 3.6

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

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Step 3.6.1

Cancel the common factor.

$\frac{9 {\cos (θ)}^{2} \cdot 1}{\sin (θ) {\cos (θ)}^{2}}$

Step 3.6.2

Rewrite the expression.

$\frac{9 \cdot 1}{\sin (θ)}$

Step 3.7

Multiply $9$ by $1$ .

$\frac{9}{\sin (θ)}$

Step 4

Rewrite $\frac{9}{\sin (θ)}$ as $9 \tan (θ) \cos (θ) \csc^{2} (θ)$ .

$9 \tan (θ) \cos (θ) \csc^{2} (θ)$

Step 5

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

$9 (\frac{\cot^{2} (θ)}{\csc (θ)}) \cdot \sec^{2} (θ) = 9 \tan (θ) \cos (θ) \csc^{2} (θ)$ is an identity