Finite Math Examples

$y = \frac{\sqrt{1 + \sin (x)}}{\sqrt{1 - \sin (x)}}$

Step 1

Simplify $\frac{\sqrt{1 + \sin (x)}}{\sqrt{1 - \sin (x)}}$ .

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Step 1.1

Multiply $\frac{\sqrt{1 + \sin (x)}}{\sqrt{1 - \sin (x)}}$ by $\frac{\sqrt{1 - \sin (x)}}{\sqrt{1 - \sin (x)}}$ .

$y = \frac{\sqrt{1 + \sin (x)}}{\sqrt{1 - \sin (x)}} \cdot \frac{\sqrt{1 - \sin (x)}}{\sqrt{1 - \sin (x)}}$

Step 1.2

Combine and simplify the denominator.

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Step 1.2.1

Multiply $\frac{\sqrt{1 + \sin (x)}}{\sqrt{1 - \sin (x)}}$ by $\frac{\sqrt{1 - \sin (x)}}{\sqrt{1 - \sin (x)}}$ .

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{\sqrt{1 - \sin (x)} \sqrt{1 - \sin (x)}}$

Step 1.2.2

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

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{{\sqrt{1 - \sin (x)}}^{1} \sqrt{1 - \sin (x)}}$

Step 1.2.3

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

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{{\sqrt{1 - \sin (x)}}^{1} {\sqrt{1 - \sin (x)}}^{1}}$

Step 1.2.4

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

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{{\sqrt{1 - \sin (x)}}^{1 + 1}}$

Step 1.2.5

Add $1$ and $1$ .

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

Step 1.2.6

Rewrite ${\sqrt{1 - \sin (x)}}^{2}$ as $1 - \sin (x)$ .

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Step 1.2.6.1

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

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

Step 1.2.6.2

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

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

Step 1.2.6.3

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

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

Step 1.2.6.4

Cancel the common factor of $2$ .

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Step 1.2.6.4.1

Cancel the common factor.

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

Step 1.2.6.4.2

Rewrite the expression.

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{{(1 - \sin (x))}^{1}}$

Step 1.2.6.5

Simplify.

$y = \frac{\sqrt{1 + \sin (x)} \sqrt{1 - \sin (x)}}{1 - \sin (x)}$

Step 1.3

Combine using the product rule for radicals.

$y = \frac{\sqrt{(1 + \sin (x)) (1 - \sin (x))}}{1 - \sin (x)}$