因式分解 sin^8(x)-cos^8(x)

解答

因式分解

sin^{8} (x) - cos^{8} (x)

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

求解步骤

sin^{8} (x) - cos^{8} (x)

sin^{8} (x) = (sin^{4} (x))^{2}

cos^{8} (x) = (cos^{4} (x))^{2}

= (sin^{4} (x))^{2} - (cos^{4} (x))^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

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

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

因式分解

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

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

s im pl i f y (x + 2)^{3} + (x - 2)^{3}

f (x) = 5 x^{5} - 3

f a c t or x^{2} y^{2} + 1 - x^{2} - y^{2}

s im pl i f y \frac{\frac{x ^{3}}{y ^{2}}}{\frac{x ^{6}}{y ^{5}}}

s im pl i f y cos (18 0^{\circ}) - x