Trigonometry Examples

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

Step 1

Simplify terms.

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Step 1.1

Simplify each term.

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Step 1.1.1

Rewrite $\cot (x)$ in terms of sines and cosines.

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

Step 1.1.2

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

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

Step 1.2

Simplify terms.

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Step 1.2.1

Combine the numerators over the common denominator.

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

Step 1.2.2

Reorder $\cos^{2} (x)$ and $- 1$ .

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

Step 1.2.3

Rewrite $- 1$ as $- 1 (1)$ .

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

Step 1.2.4

Factor $- 1$ out of $\cos^{2} (x)$ .

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

Step 1.2.5

Factor $- 1$ out of $- 1 (1) - 1 (- \cos^{2} (x))$ .

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

Step 1.2.6

Rewrite $- 1 (1 - \cos^{2} (x))$ as $- (1 - \cos^{2} (x))$ .

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

Step 2

Apply pythagorean identity.

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

Step 3

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

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Step 3.1

Cancel the common factor.

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

Step 3.2

Divide $- 1$ by $1$ .

$- 1$