Trigonometry Examples

$\frac{{(8 \cos (x))}^{2}}{\sqrt{64 - x^{2}}}$

Step 1

Simplify the numerator.

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Step 1.1

Apply the product rule to $8 \cos (x)$ .

$\frac{8^{2} \cos^{2} (x)}{\sqrt{64 - x^{2}}}$

Step 1.2

Raise $8$ to the power of $2$ .

$\frac{64 \cos^{2} (x)}{\sqrt{64 - x^{2}}}$

Step 2

Simplify the denominator.

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Step 2.1

Rewrite $64$ as $8^{2}$ .

$\frac{64 \cos^{2} (x)}{\sqrt{8^{2} - x^{2}}}$

Step 2.2

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 8$ and $b = x$ .

$\frac{64 \cos^{2} (x)}{\sqrt{(8 + x) (8 - x)}}$

Step 3

Multiply $\frac{64 \cos^{2} (x)}{\sqrt{(8 + x) (8 - x)}}$ by $\frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$ .

$\frac{64 \cos^{2} (x)}{\sqrt{(8 + x) (8 - x)}} \cdot \frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$

Step 4

Combine and simplify the denominator.

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Step 4.1

Multiply $\frac{64 \cos^{2} (x)}{\sqrt{(8 + x) (8 - x)}}$ by $\frac{\sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)}}$ .

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{\sqrt{(8 + x) (8 - x)} \sqrt{(8 + x) (8 - x)}}$

Step 4.2

Raise $\sqrt{(8 + x) (8 - x)}$ to the power of $1$ .

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1} \sqrt{(8 + x) (8 - x)}}$

Step 4.3

Raise $\sqrt{(8 + x) (8 - x)}$ to the power of $1$ .

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1} {\sqrt{(8 + x) (8 - x)}}^{1}}$

Step 4.4

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

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{1 + 1}}$

Step 4.5

Add $1$ and $1$ .

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{\sqrt{(8 + x) (8 - x)}}^{2}}$

Step 4.6

Rewrite ${\sqrt{(8 + x) (8 - x)}}^{2}$ as $(8 + x) (8 - x)$ .

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Step 4.6.1

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{(8 + x) (8 - x)}$ as ${((8 + x) (8 - x))}^{\frac{1}{2}}$ .

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{({((8 + x) (8 - x))}^{\frac{1}{2}})}^{2}}$

Step 4.6.2

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

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{1}{2} \cdot 2}}$

Step 4.6.3

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

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{2}{2}}}$

Step 4.6.4

Cancel the common factor of $2$ .

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Step 4.6.4.1

Cancel the common factor.

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{\frac{2}{2}}}$

Step 4.6.4.2

Rewrite the expression.

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{{((8 + x) (8 - x))}^{1}}$

Step 4.6.5

Simplify.

$\frac{64 \cos^{2} (x) \sqrt{(8 + x) (8 - x)}}{(8 + x) (8 - x)}$