簡約 (sin(x)-sin(y))^2+(cos(x)-cos(y))^2-2

解

簡約

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

- 2 cos (x - y)

解答ステップ

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

拡張

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

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

拡張

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

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

ピタゴラスの公式を使用する:

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

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

数を足す:

1 + 1 - 2 = 0

= - 2 cos (x) cos (y) - 2 sin (x) sin (y)

角の差の公式を使用する:

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

= - 2 cos (x - y)

共通項をくくり出す

2

= - 2 cos (x - y)