因数 sin^8(x)-cos^8(x)

解

因数

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))

解答ステップ

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}

2乗の差の公式を適用する:

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))

因数

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))