Trigonometry Examples

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

Step 1

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

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

Step 2

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

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

Step 3

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

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

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

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

Step 5

Simplify the numerator.

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Step 5.1

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

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

Step 5.2

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

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

Step 5.3

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

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

Step 5.4

Add $1$ and $1$ .

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

Step 6

Rewrite $\sqrt{\frac{\cos^{2} (x)}{\sin (x)}}$ as $\frac{\sqrt{\cos^{2} (x)}}{\sqrt{\sin (x)}}$ .

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

Step 7

Pull terms out from under the radical, assuming positive real numbers.

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

Step 8

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

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

Step 9

Combine and simplify the denominator.

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Step 9.1

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

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

Step 9.2

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

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

Step 9.3

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

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

Step 9.4

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

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

Step 9.5

Add $1$ and $1$ .

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

Step 9.6

Rewrite ${\sqrt{\sin (x)}}^{2}$ as $\sin (x)$ .

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Step 9.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{\sin (x)}$ as ${\sin (x)}^{\frac{1}{2}}$ .

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

Step 9.6.2

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Step 9.6.3

Combine $\frac{1}{2}$ and $2$ .

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

Step 9.6.4

Cancel the common factor of $2$ .

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Step 9.6.4.1

Cancel the common factor.

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

Step 9.6.4.2

Rewrite the expression.

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

Step 9.6.5

Simplify.

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

Step 10

Separate fractions.

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

Step 11

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

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

Step 12

Divide $\sqrt{\sin (x)}$ by $1$ .

$\sqrt{\sin (x)} \cot (x)$