Trigonometry Examples

$1 - \cos^{2} (x) - 2 \cos (x) - 2$

Step 1

Subtract $2$ from $1$ .

$- 1 - \cos^{2} (x) - 2 \cos (x)$

Step 2

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

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Step 2.1

Rewrite $- 1$ as $- 1 (1)$ .

$- 1 (1) - \cos^{2} (x) - 2 \cos (x)$

Step 2.2

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

$- 1 (1) - (\cos^{2} (x)) - 2 \cos (x)$

Step 2.3

Factor $- 1$ out of $- 2 \cos (x)$ .

$- 1 (1) - (\cos^{2} (x)) - (2 \cos (x))$

Step 2.4

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

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

Step 2.5

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

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

Step 2.6

Rewrite $- 1 (1 + \cos^{2} (x) + 2 \cos (x))$ as $- (1 + \cos^{2} (x) + 2 \cos (x))$ .

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

Step 3

Factor using the perfect square rule.

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Step 3.1

Rearrange terms.

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

Step 3.2

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

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

Step 3.3

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

$2 \cos (x) = 2 \cdot 1 \cdot \cos (x)$

Step 3.4

Rewrite the polynomial.

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

Step 3.5

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

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