Trigonometry Examples

$9 \cos^{2} (x) + 18 \cos (x) + 9$

Step 1

Factor $9$ out of $9 \cos^{2} (x) + 18 \cos (x) + 9$ .

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Step 1.1

Factor $9$ out of $9 \cos^{2} (x)$ .

$9 (\cos^{2} (x)) + 18 \cos (x) + 9$

Step 1.2

Factor $9$ out of $18 \cos (x)$ .

$9 (\cos^{2} (x)) + 9 (2 \cos (x)) + 9$

Step 1.3

Factor $9$ out of $9$ .

$9 (\cos^{2} (x)) + 9 (2 \cos (x)) + 9 (1)$

Step 1.4

Factor $9$ out of $9 (\cos^{2} (x)) + 9 (2 \cos (x))$ .

$9 (\cos^{2} (x) + 2 \cos (x)) + 9 (1)$

Step 1.5

Factor $9$ out of $9 (\cos^{2} (x) + 2 \cos (x)) + 9 (1)$ .

$9 (\cos^{2} (x) + 2 \cos (x) + 1)$

Step 2

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

$9 (u^{2} + 2 u + 1)$

Step 3

Factor using the perfect square rule.

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Step 3.1

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

$9 (u^{2} + 2 u + 1^{2})$

Step 3.2

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

$2 u = 2 \cdot u \cdot 1$

Step 3.3

Rewrite the polynomial.

$9 (u^{2} + 2 \cdot u \cdot 1 + 1^{2})$

Step 3.4

Factor using the perfect square trinomial rule $a^{2} + 2 a b + b^{2} = {(a + b)}^{2}$ , where $a = u$ and $b = 1$ .

$9 {(u + 1)}^{2}$

Step 4

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

$9 {(\cos (x) + 1)}^{2}$