Trigonometry Examples

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

Step 1

Separate fractions.

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

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

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

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

Step 5.2

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

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

Step 6

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

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

Step 7

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

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

Step 8

Simplify the numerator.

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Step 8.1

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

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

Step 8.2

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

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

Step 8.3

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

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

Step 8.4

Add $1$ and $1$ .

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

Step 9

Combine fractions.

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Step 9.1

Combine.

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

Step 9.2

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

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

Step 10

Simplify the denominator.

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Step 10.1

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

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

Step 10.2

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

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

Step 10.3

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

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

Step 10.4

Add $1$ and $1$ .

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

Step 11

Convert from $\frac{\cos^{2} (x)}{\sin^{2} (x)}$ to $\cot^{2} (x)$ .

$\cot^{2} (x)$