Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

Apply the product rule to $\frac{1}{\sin (x)}$ .

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

Step 9

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

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Step 9.1

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

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

Step 9.2

Cancel the common factor.

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

Step 9.3

Rewrite the expression.

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

Step 10

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

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

Step 11

Separate fractions.

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

Step 12

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

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

Step 13

Separate fractions.

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

Step 14

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

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

Step 15

Divide $1^{2}$ by $1$ .

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

Step 16

One to any power is one.

$\cot (x) - (1 \sec (x) \csc (x))$

Step 17

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

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