Trigonometry Examples

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

Step 1

Apply pythagorean identity.

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

Step 2

Simplify the denominator.

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Step 2.1

Since $\sin (- x)$ is an odd function, rewrite $\sin (- x)$ as $- \sin (x)$ .

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

Step 2.2

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

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

Step 2.3

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

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

Step 2.4

Reduce the expression by cancelling the common factors.

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Step 2.4.1

Reduce the expression $\frac{\cos (x) \sin (x)}{\sin (x)}$ by cancelling the common factors.

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Step 2.4.1.1

Cancel the common factor.

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

Step 2.4.1.2

Rewrite the expression.

$\frac{1}{- \frac{\cos (x)}{1}}$

Step 2.4.2

Divide $\cos (x)$ by $1$ .

$\frac{1}{- \cos (x)}$

Step 3

Cancel the common factor of $1$ and $- 1$ .

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Step 3.1

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

$\frac{- 1 (- 1)}{- \cos (x)}$

Step 3.2

Move the negative in front of the fraction.

$- \frac{1}{\cos (x)}$

Step 4

Convert from $\frac{1}{\cos (x)}$ to $\sec (x)$ .

$- \sec (x)$