Precalculus Examples

$\tan (\arcsin (\frac{x}{2}))$

Step 1

Draw a triangle in the plane with vertices $(\sqrt{1^{2} - {(\frac{x}{2})}^{2}}, \frac{x}{2})$ , $(\sqrt{1^{2} - {(\frac{x}{2})}^{2}}, 0)$ , and the origin. Then $\arcsin (\frac{x}{2})$ is the angle between the positive x-axis and the ray beginning at the origin and passing through $(\sqrt{1^{2} - {(\frac{x}{2})}^{2}}, \frac{x}{2})$ . Therefore, $\tan (\arcsin (\frac{x}{2}))$ is $\frac{\frac{x}{2}}{\sqrt{1 - {(\frac{x}{2})}^{2}}}$ .

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

Step 2

Multiply the numerator by the reciprocal of the denominator.

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

Step 3

Combine fractions.

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Step 3.1

Combine.

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

Step 3.2

Multiply $x$ by $1$ .

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

Step 4

Simplify the denominator.

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Step 4.1

Rewrite $1$ as $1^{2}$ .

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

Step 4.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \frac{x}{2}$ .

$\frac{x}{2 \sqrt{(1 + \frac{x}{2}) (1 - \frac{x}{2})}}$

Step 4.3

Simplify.

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Step 4.3.1

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

$\frac{x}{2 \sqrt{(\frac{2}{2} + \frac{x}{2}) (1 - \frac{x}{2})}}$

Step 4.3.2

Combine the numerators over the common denominator.

$\frac{x}{2 \sqrt{\frac{2 + x}{2} (1 - \frac{x}{2})}}$

Step 4.3.3

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

$\frac{x}{2 \sqrt{\frac{2 + x}{2} (\frac{2}{2} - \frac{x}{2})}}$

Step 4.3.4

Combine the numerators over the common denominator.

$\frac{x}{2 \sqrt{\frac{2 + x}{2} \cdot \frac{2 - x}{2}}}$

Step 4.4

Multiply $\frac{2 + x}{2}$ by $\frac{2 - x}{2}$ .

$\frac{x}{2 \sqrt{\frac{(2 + x) (2 - x)}{2 \cdot 2}}}$

Step 4.5

Multiply $2$ by $2$ .

$\frac{x}{2 \sqrt{\frac{(2 + x) (2 - x)}{4}}}$

Step 4.6

Rewrite $\frac{(2 + x) (2 - x)}{4}$ as ${(\frac{1}{2})}^{2} ((2 + x) (2 - x))$ .

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Step 4.6.1

Factor the perfect power $1^{2}$ out of $(2 + x) (2 - x)$ .

$\frac{x}{2 \sqrt{\frac{1^{2} ((2 + x) (2 - x))}{4}}}$

Step 4.6.2

Factor the perfect power $2^{2}$ out of $4$ .

$\frac{x}{2 \sqrt{\frac{1^{2} ((2 + x) (2 - x))}{2^{2} \cdot 1}}}$

Step 4.6.3

Rearrange the fraction $\frac{1^{2} ((2 + x) (2 - x))}{2^{2} \cdot 1}$ .

$\frac{x}{2 \sqrt{{(\frac{1}{2})}^{2} ((2 + x) (2 - x))}}$

Step 4.7

Pull terms out from under the radical.

$\frac{x}{2 (\frac{1}{2} \sqrt{(2 + x) (2 - x)})}$

Step 4.8

Combine $\frac{1}{2}$ and $\sqrt{(2 + x) (2 - x)}$ .

$\frac{x}{2 \frac{\sqrt{(2 + x) (2 - x)}}{2}}$

Step 5

Combine $2$ and $\frac{\sqrt{(2 + x) (2 - x)}}{2}$ .

$\frac{x}{\frac{2 \sqrt{(2 + x) (2 - x)}}{2}}$

Step 6

Reduce the expression by cancelling the common factors.

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Step 6.1

Reduce the expression $\frac{2 \sqrt{(2 + x) (2 - x)}}{2}$ by cancelling the common factors.

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Step 6.1.1

Cancel the common factor.

$\frac{x}{\frac{2 \sqrt{(2 + x) (2 - x)}}{2}}$

Step 6.1.2

Rewrite the expression.

$\frac{x}{\frac{\sqrt{(2 + x) (2 - x)}}{1}}$

Step 6.2

Divide $\sqrt{(2 + x) (2 - x)}$ by $1$ .

$\frac{x}{\sqrt{(2 + x) (2 - x)}}$

Step 7

Multiply $\frac{x}{\sqrt{(2 + x) (2 - x)}}$ by $\frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}$ .

$\frac{x}{\sqrt{(2 + x) (2 - x)}} \cdot \frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}$

Step 8

Combine and simplify the denominator.

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Step 8.1

Multiply $\frac{x}{\sqrt{(2 + x) (2 - x)}}$ by $\frac{\sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)}}$ .

$\frac{x \sqrt{(2 + x) (2 - x)}}{\sqrt{(2 + x) (2 - x)} \sqrt{(2 + x) (2 - x)}}$

Step 8.2

Raise $\sqrt{(2 + x) (2 - x)}$ to the power of $1$ .

$\frac{x \sqrt{(2 + x) (2 - x)}}{{\sqrt{(2 + x) (2 - x)}}^{1} \sqrt{(2 + x) (2 - x)}}$

Step 8.3

Raise $\sqrt{(2 + x) (2 - x)}$ to the power of $1$ .

$\frac{x \sqrt{(2 + x) (2 - x)}}{{\sqrt{(2 + x) (2 - x)}}^{1} {\sqrt{(2 + x) (2 - x)}}^{1}}$

Step 8.4

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

$\frac{x \sqrt{(2 + x) (2 - x)}}{{\sqrt{(2 + x) (2 - x)}}^{1 + 1}}$

Step 8.5

Add $1$ and $1$ .

$\frac{x \sqrt{(2 + x) (2 - x)}}{{\sqrt{(2 + x) (2 - x)}}^{2}}$

Step 8.6

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

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Step 8.6.1

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

$\frac{x \sqrt{(2 + x) (2 - x)}}{{({((2 + x) (2 - x))}^{\frac{1}{2}})}^{2}}$

Step 8.6.2

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

$\frac{x \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{1}{2} \cdot 2}}$

Step 8.6.3

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

$\frac{x \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{2}{2}}}$

Step 8.6.4

Cancel the common factor of $2$ .

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Step 8.6.4.1

Cancel the common factor.

$\frac{x \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{\frac{2}{2}}}$

Step 8.6.4.2

Rewrite the expression.

$\frac{x \sqrt{(2 + x) (2 - x)}}{{((2 + x) (2 - x))}^{1}}$

Step 8.6.5

Simplify.

$\frac{x \sqrt{(2 + x) (2 - x)}}{(2 + x) (2 - x)}$