Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

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

Step 9

Simplify terms.

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Step 9.1

Combine.

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

Step 9.2

Cancel the common factor of $\sin^{2} (x)$ and $\sin (x)$ .

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Step 9.2.1

Factor $\sin (x)$ out of $1 \sin^{2} (x)$ .

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

Step 9.2.2

Cancel the common factors.

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Step 9.2.2.1

Factor $\sin (x)$ out of $\sin (x) \cos (x)$ .

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

Step 9.2.2.2

Cancel the common factor.

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

Step 9.2.2.3

Rewrite the expression.

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

Step 9.3

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

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

Step 10

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

$\tan (x)$