展開する (f'(t)sin(t)+f(t)cos(t))^2

解

展開する

(f^{'} (t) sin (t) + f (t) cos (t))^{2}

sin^{2} (t) (f' (t))^{2} + 2 sin (t) cos (t) f' (t) f (t) + cos^{2} (t) f (t)^{2}

解答ステップ

(f^{'} (t) sin (t) + f (t) cos (t))^{2}

= (sin (t) f^{^{'}} (t) + cos (t) f (t))^{2}

完全平方式を適用する:

(a + b)^{2} = a^{2} + 2 ab + b^{2}

a = f^{'} (t) sin (t), b = f (t) cos (t)

= (f^{^{'}} (t) sin (t))^{2} + 2 f^{^{'}} (t) sin (t) f (t) cos (t) + (f (t) cos (t))^{2}

簡素化

(f^{'} (t) sin (t))^{2} + 2 f^{'} (t) sin (t) f (t) cos (t) + (f (t) cos (t))^{2} : sin^{2} (t) (f^{'} (t))^{2} + 2 sin (t) cos (t) f^{'} (t) f (t) + cos^{2} (t) f (t)^{2}

= sin^{2} (t) (f^{'} (t))^{2} + 2 sin (t) cos (t) f^{'} (t) f (t) + cos^{2} (t) f (t)^{2}