Trigonometry Examples

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

Step 1

Simplify the expression.

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Step 1.1

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

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

Step 1.2

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

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

Step 2

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

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

Step 3

Rearrange terms.

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

Step 4

Apply pythagorean identity.

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

Step 5

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

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

Step 6

Apply the cosine double-angle identity.

$\cos (2 t)$