Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

Simplify the numerator.

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Step 3.1

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 3.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 3.4

Add $1$ and $1$ .

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

Step 4

Multiply the numerator by the reciprocal of the denominator.

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

Step 5

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

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

Step 6

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

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

Step 7

Separate fractions.

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

Step 8

Convert from $\frac{\cos (x)}{\sin (x)}$ to $\cot (x)$ .

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

Step 9

Combine $\frac{\cos (x)}{1 - \sin (x)}$ and $\cot (x)$ .

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