Trigonometry Examples

$\frac{\cos^{2} (x) + 8 \cos (x) + 16}{\cos (x) + 4}$

Step 1

Simplify the numerator.

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Step 1.1

Let $u = \cos (x)$ . Substitute $u$ for all occurrences of $\cos (x)$ .

$\frac{u^{2} + 8 u + 16}{\cos (x) + 4}$

Step 1.2

Factor using the perfect square rule.

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Step 1.2.1

Rewrite $16$ as $4^{2}$ .

$\frac{u^{2} + 8 u + 4^{2}}{\cos (x) + 4}$

Step 1.2.2

Check that the middle term is two times the product of the numbers being squared in the first term and third term.

$8 u = 2 \cdot u \cdot 4$

Step 1.2.3

Rewrite the polynomial.

$\frac{u^{2} + 2 \cdot u \cdot 4 + 4^{2}}{\cos (x) + 4}$

Step 1.2.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 4$ .

$\frac{{(u + 4)}^{2}}{\cos (x) + 4}$

Step 1.3

Replace all occurrences of $u$ with $\cos (x)$ .

$\frac{{(\cos (x) + 4)}^{2}}{\cos (x) + 4}$

Step 2

Cancel the common factor of ${(\cos (x) + 4)}^{2}$ and $\cos (x) + 4$ .

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Step 2.1

Factor $\cos (x) + 4$ out of ${(\cos (x) + 4)}^{2}$ .

$\frac{(\cos (x) + 4) (\cos (x) + 4)}{\cos (x) + 4}$

Step 2.2

Cancel the common factors.

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Step 2.2.1

Multiply by $1$ .

$\frac{(\cos (x) + 4) (\cos (x) + 4)}{(\cos (x) + 4) \cdot 1}$

Step 2.2.2

Cancel the common factor.

$\frac{(\cos (x) + 4) (\cos (x) + 4)}{(\cos (x) + 4) \cdot 1}$

Step 2.2.3

Rewrite the expression.

$\frac{\cos (x) + 4}{1}$

Step 2.2.4

Divide $\cos (x) + 4$ by $1$ .

$\cos (x) + 4$