Trigonometry Examples

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

Step 1

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

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

Step 2

Separate fractions.

$\frac{\sin (x)}{\tan^{2} (x)} \cdot \frac{\sin (x)}{\csc (x)}$

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.2

Raise $\sin (x)$ to the power of $1$ .

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

Step 5.3

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

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

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

Step 7

Separate fractions.

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

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

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Step 11.1

Cancel the common factor.

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

Step 11.2

Rewrite the expression.

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

Step 12

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

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

Step 13

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

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

Step 14

Write $1$ as a fraction with denominator $1$ .

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

Step 15

Simplify.

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Step 15.1

Rewrite the expression.

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

Step 15.2

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

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

Step 16

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

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Step 16.1

Combine $\frac{\cos (x)}{\sin (x)}$ and $\cos (x)$ .

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

Step 16.2

Raise $\cos (x)$ to the power of $1$ .

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

Step 16.3

Raise $\cos (x)$ to the power of $1$ .

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

Step 16.4

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

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

Step 16.5

Add $1$ and $1$ .

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

Step 17

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

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Step 17.1

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

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

Step 17.2

Cancel the common factor.

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

Step 17.3

Rewrite the expression.

$\cos^{2} (x) \sin (x)$