Simplify csc(x)-(cot(x)^2)/(csc(x))

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

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

Multiply by the reciprocal of the fraction to divide by $\frac{1}{\sin (x)}$ .

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

Step 4

Multiply $\sin (x)$ by $1$ .

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

Step 5

Divide $\cot^{2} (x)$ by $1$ .

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

Step 6

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

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

Step 7

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

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

Step 8

Cancel the common factor of $\sin (x)$ .

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

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

Cancel the common factor.

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

Rewrite the expression.

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

Step 9

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

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

Step 10

Separate fractions.

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

Step 11

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

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

Step 12

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

$\csc (x) - \cos (x) \cot (x)$