Factor sin(x)^6+cos(x)^6

$\sin^{6} (x) + \cos^{6} (x)$

Rewrite $\sin^{6} (x)$ as ${(\sin^{2} (x))}^{3}$ .

${(\sin^{2} (x))}^{3} + \cos^{6} (x)$

Rewrite $\cos^{6} (x)$ as ${(\cos^{2} (x))}^{3}$ .

${(\sin^{2} (x))}^{3} + {(\cos^{2} (x))}^{3}$

Since both terms are perfect cubes, factor using the sum of cubes formula, $a^{3} + b^{3} = (a + b) (a^{2} - a b + b^{2})$ where $a = \sin^{2} (x)$ and $b = \cos^{2} (x)$ .

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

Simplify.

Apply pythagorean identity.

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

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

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

Multiply the exponents in ${(\sin^{2} (x))}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

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

Multiply the exponents in ${(\cos^{2} (x))}^{2}$ .

Apply the power rule and multiply exponents, ${(a^{m})}^{n} = a^{m n}$ .

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

Multiply $2$ by $2$ .

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

Factor sin(x)^6+cos(x)^6

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