Trigonometry Examples

$\frac{\csc^{2} (x) - 1}{\csc^{2} (x)} = \cos^{2} (x)$

Step 1

Start on the left side.

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

Step 2

Apply pythagorean identity.

$\frac{\cot^{2} (x)}{\csc^{2} (x)}$

Step 3

Convert to sines and cosines.

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Step 3.1

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

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Multiply the numerator by the reciprocal of the denominator.

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

Step 4.2

Combine.

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

Step 4.3

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

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Step 4.3.1

Cancel the common factor.

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

Step 4.3.2

Rewrite the expression.

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

Step 4.4

One to any power is one.

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

Step 4.5

Divide ${\cos (x)}^{2}$ by $1$ .

$\cos^{2} (x)$

Step 5

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

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