Trigonometry Examples

$1 - 2 \cos^{2} (x) + \cos^{4} (x)$

Step 1

Factor using the perfect square rule.

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Step 1.1

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

$1^{2} - 2 \cos^{2} (x) + \cos^{4} (x)$

Step 1.2

Rewrite $\cos^{4} (x)$ as ${(\cos^{2} (x))}^{2}$ .

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

Step 1.3

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

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

Step 1.4

Rewrite the polynomial.

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

Step 1.5

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

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

Step 2

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

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

Step 3

Since both terms are perfect squares, factor using the difference of squares formula, $a^{2} - b^{2} = (a + b) (a - b)$ where $a = 1$ and $b = \cos (x)$ .

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

Step 4

Apply the product rule to $(1 + \cos (x)) (1 - \cos (x))$ .

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