Trigonometry Examples

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

Step 1

Start on the left side.

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

Step 2

Apply Pythagorean identity in reverse.

$1 + \cot^{2} (x) - \cos^{2} (x) \csc^{2} (x)$

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.

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

Step 3.2

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

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

Step 3.3

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

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

Step 3.4

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

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

Step 4

Simplify.

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Step 4.1

Simplify each term.

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Step 4.1.1

One to any power is one.

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

Step 4.1.2

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

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

Step 4.2

Combine the numerators over the common denominator.

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

Step 4.3

Subtract ${\cos (x)}^{2}$ from ${\cos (x)}^{2}$ .

$1 + \frac{0}{{\sin (x)}^{2}}$

Step 4.4

Divide $0$ by ${\sin (x)}^{2}$ .

$1 + 0$

Step 4.5

Add $1$ and $0$ .

$1$

Step 5

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

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