展開する (sin(x)+cos(x))^4

解

展開する

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

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

解答ステップ

(sin (x) + cos (x))^{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 = sin (x), b = cos (x)

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

総和を展開する

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

簡素化

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

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