Algebra Examples

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

Step 1

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

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

Step 2

Separate fractions.

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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Step 6.1

Cancel the common factor.

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

Step 6.2

Rewrite the expression.

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Separate fractions.

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

Step 12

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

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

Step 13

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

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

Step 14

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

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

Step 15

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

$\tan (x) \sec (x) - \tan^{2} (x) \sin (x)$