簡約 (sin(a)+cos(b))^2+(cos(a)+sin(b))^2

解

簡約

(sin (a) + cos (b))^{2} + (cos (a) + sin (b))^{2}

2 sin (a + b) + 2

解答ステップ

(sin (a) + cos (b))^{2} + (cos (a) + sin (b))^{2}

拡張

(sin (a) + cos (b))^{2} : sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b)

= sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b) + (cos (a) + sin (b))^{2}

拡張

(cos (a) + sin (b))^{2} : cos^{2} (a) + 2 cos (a) sin (b) + sin^{2} (b)

= sin^{2} (a) + 2 cos (b) sin (a) + cos^{2} (b) + cos^{2} (a) + 2 cos (a) sin (b) + sin^{2} (b)

ピタゴラスの公式を使用する:

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

= 2 cos (a) sin (b) + 2 cos (b) sin (a) + 1 + 1

数を足す:

1 + 1 = 2

= 2 cos (a) sin (b) + 2 cos (b) sin (a) + 2

角の和の公式を使用する:

sin (s) cos (t) + cos (s) sin (t) = sin (s + t)

= 2 (1 + sin (a + b))

因数

2 cos (a) sin (b) + 2 cos (b) sin (a) + 2 : 2 (cos (a) sin (b) + cos (b) sin (a) + 1)

= 2 (sin (a + b) + 1)

拡張

2 (sin (a + b) + 1) : 2 sin (a + b) + 2

= 2 sin (a + b) + 2