What is sin^4(x)-cos^4(x) ?

The solution to sin^4(x)-cos^4(x) is -cos(2x)

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simplify sin^4(x)-cos^4(x)

simplify

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

- cos (2 x)

Solution steps

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

Factor

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

= (- cos (x) + sin (x)) (cos (x) + sin (x)) (cos^{2} (x) + sin^{2} (x))

Rewrite using trig identities

= (- cos (x) + sin (x)) (cos (x) + sin (x))

Expand

(- cos (x) + sin (x)) (cos (x) + sin (x)) : sin^{2} (x) - cos^{2} (x)

= - cos^{2} (x) + sin^{2} (x)

Rewrite using trig identities

= - cos (2 x)

\frac{2 x + 1}{x} + \frac{3}{x ^{2}}

\frac{1}{8} \cdot 1 0^{- 2}

3^{\frac{3}{2}} + 3 i

2 \frac{1}{2} + 1 \frac{1}{5} - 3 \frac{3}{10}

cos^{2} (x) - 1