Trigonometry Examples

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1} = \sec^{2} (θ)$

Step 1

Start on the left side.

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1}$

Step 2

Apply pythagorean identity.

$\frac{\csc^{2} (θ)}{\cot^{2} (θ)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Combine.

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

Step 4.3

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

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

$\frac{1^{2}}{{\cos (θ)}^{2}}$

Step 4.4

One to any power is one.

$\frac{1}{\cos^{2} (θ)}$

Step 5

Rewrite $\frac{1}{\cos^{2} (θ)}$ as $\sec^{2} (θ)$ .

$\sec^{2} (θ)$

Step 6

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

$\frac{\csc^{2} (θ)}{\csc^{2} (θ) - 1} = \sec^{2} (θ)$ is an identity