Trigonometry Examples

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

Step 1

Simplify the numerator.

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Step 1.1

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

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

Step 1.2

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

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

Step 1.3

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

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

Step 1.4

Reduce the expression by cancelling the common factors.

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Step 1.4.1

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

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Step 1.4.1.1

Cancel the common factor.

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

Step 1.4.1.2

Rewrite the expression.

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

Step 1.4.2

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

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

Step 2

Move the negative in front of the fraction.

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

Step 3

Rewrite $\frac{\cos (x)}{\sin (\frac{π}{2} - x)}$ as a product.

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

Step 4

Write $\cos (x)$ as a fraction with denominator $1$ .

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

Convert from $\frac{1}{\sin (\frac{π}{2} - x)}$ to $\csc (\frac{π}{2} - x)$ .

$- (\cos (x) \csc (\frac{π}{2} - x))$

$- \cos (x) \csc (\frac{π}{2} - x)$