化简 (sin(x)+cos(x))^2(sin(x)-cos(x))^2

解答

化简

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

cos^{2} (2 x)

求解步骤

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

化简

(sin (x) + cos (x))^{2} : sin (2 x) + 1

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

化简

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

= (sin (2 x) + 1) (- sin (2 x) + 1)

乘开

(1 + sin (2 x)) (1 - sin (2 x)) : 1 - sin^{2} (2 x)

= 1 - sin^{2} (2 x)

使用毕达哥拉斯恒等式:

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

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

= cos^{2} (2 x)

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