Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

$\frac{\cot (x)}{1} \cdot \frac{\tan (x)}{\frac{1}{\sin (x)}}$

Step 3

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

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

Step 4

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

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

Step 5

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

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

Step 6

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

$\cot (x) (\sin (x) \tan (x))$

Step 7

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 7.1

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

$\frac{\cos (x)}{\sin (x)} \sin (x) \tan (x)$

Step 7.2

Cancel the common factors.

$\cos (x) \tan (x)$

Step 8

Rewrite in terms of sines and cosines, then cancel the common factors.

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Step 8.1

Reorder $\cos (x)$ and $\tan (x)$ .

$\tan (x) \cos (x)$

Step 8.2

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

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

Step 8.3

Cancel the common factors.

$\sin (x)$