Trigonometry Examples

$\cos^{4} (x) - \sin^{4} (x)$

Step 1

The polynomial cannot be factored using the grouping method. Try a different method, or if you aren't sure, choose Factor.

The polynomial cannot be factored using the grouping method.

Step 2

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

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

Step 3

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

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

Step 4

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

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

Step 5

Simplify.

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Step 5.1

Rearrange terms.

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

Step 5.2

Apply pythagorean identity.

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

Step 5.3

Multiply $\cos^{2} (x) - \sin^{2} (x)$ by $1$ .

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

Step 5.4

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

$(\cos (x) + \sin (x)) (\cos (x) - \sin (x))$