Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

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

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Step 4.1

Cancel the common factor.

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

Step 4.2

Rewrite the expression.

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

Step 5

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

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

Step 6

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

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

Step 7

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

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

Step 8

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

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

Step 9

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

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

Step 10

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

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Step 10.1

Cancel the common factor.

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

Step 10.2

Rewrite the expression.

$\sec (x) - \cos (x)$