Precalculus Examples

$\tan (\frac{a}{2})$

Step 1

Apply the tangent half-angle identity.

$\pm \sqrt{\frac{1 - \cos (a)}{1 + \cos (a)}}$

Step 2

Rewrite $\sqrt{\frac{1 - \cos (a)}{1 + \cos (a)}}$ as $\frac{\sqrt{1 - \cos (a)}}{\sqrt{1 + \cos (a)}}$ .

$\pm \frac{\sqrt{1 - \cos (a)}}{\sqrt{1 + \cos (a)}}$

Step 3

Multiply $\frac{\sqrt{1 - \cos (a)}}{\sqrt{1 + \cos (a)}}$ by $\frac{\sqrt{1 + \cos (a)}}{\sqrt{1 + \cos (a)}}$ .

$\pm \frac{\sqrt{1 - \cos (a)}}{\sqrt{1 + \cos (a)}} \cdot \frac{\sqrt{1 + \cos (a)}}{\sqrt{1 + \cos (a)}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{\sqrt{1 - \cos (a)}}{\sqrt{1 + \cos (a)}}$ by $\frac{\sqrt{1 + \cos (a)}}{\sqrt{1 + \cos (a)}}$ .

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{\sqrt{1 + \cos (a)} \sqrt{1 + \cos (a)}}$

Step 4.2

Raise $\sqrt{1 + \cos (a)}$ to the power of $1$ .

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{\sqrt{1 + \cos (a)}}^{1} \sqrt{1 + \cos (a)}}$

Step 4.3

Raise $\sqrt{1 + \cos (a)}$ to the power of $1$ .

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{\sqrt{1 + \cos (a)}}^{1} {\sqrt{1 + \cos (a)}}^{1}}$

Step 4.4

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

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{\sqrt{1 + \cos (a)}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{\sqrt{1 + \cos (a)}}^{2}}$

Step 4.6

Rewrite ${\sqrt{1 + \cos (a)}}^{2}$ as $1 + \cos (a)$ .

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Step 4.6.1

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

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{({(1 + \cos (a))}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{(1 + \cos (a))}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{(1 + \cos (a))}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{(1 + \cos (a))}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{{(1 + \cos (a))}^{1}}$

Step 4.6.5

Simplify.

$\pm \frac{\sqrt{1 - \cos (a)} \sqrt{1 + \cos (a)}}{1 + \cos (a)}$

Step 5

Combine using the product rule for radicals.

$\pm \frac{\sqrt{(1 - \cos (a)) (1 + \cos (a))}}{1 + \cos (a)}$