展開する (cos(t)+sin(t))^4

解

展開する

(cos (t) + sin (t))^{4}

cos^{4} (t) + 4 cos^{3} (t) sin (t) + 6 cos^{2} (t) sin^{2} (t) + 4 sin^{3} (t) cos (t) + sin^{4} (t)

解答ステップ

(cos (t) + sin (t))^{4}

2項定理を適用する:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = cos (t), b = sin (t)

= i = 0 \sum 4 (i 4) cos^{(4 - i)} (t) sin^{i} (t)

総和を展開する

= \frac{4 !}{0 ! ( 4 - 0 ) !} cos^{4} (t) sin^{0} (t) + \frac{4 !}{1 ! ( 4 - 1 ) !} cos^{3} (t) sin^{1} (t) + \frac{4 !}{2 ! ( 4 - 2 ) !} cos^{2} (t) sin^{2} (t) + \frac{4 !}{3 ! ( 4 - 3 ) !} cos^{1} (t) sin^{3} (t) + \frac{4 !}{4 ! ( 4 - 4 ) !} cos^{0} (t) sin^{4} (t)

簡素化

\frac{4 !}{0 ! ( 4 - 0 ) !} cos^{4} (t) sin^{0} (t) : cos^{4} (t)

簡素化

\frac{4 !}{1 ! ( 4 - 1 ) !} cos^{3} (t) sin^{1} (t) : 4 cos^{3} (t) sin (t)

簡素化

\frac{4 !}{2 ! ( 4 - 2 ) !} cos^{2} (t) sin^{2} (t) : 6 cos^{2} (t) sin^{2} (t)

簡素化

\frac{4 !}{3 ! ( 4 - 3 ) !} cos^{1} (t) sin^{3} (t) : 4 sin^{3} (t) cos (t)

簡素化

\frac{4 !}{4 ! ( 4 - 4 ) !} cos^{0} (t) sin^{4} (t) : sin^{4} (t)

= cos^{4} (t) + 4 cos^{3} (t) sin (t) + 6 cos^{2} (t) sin^{2} (t) + 4 sin^{3} (t) cos (t) + sin^{4} (t)