展開する ((1-cos(x))^3cos^2(x))/3

解

展開する

\frac{( 1 - cos ( x ) ) ^{3} cos ^{2} ( x )}{3}

\frac{cos ^{2} ( x )}{3} - cos^{3} (x) + cos^{4} (x) - \frac{cos ^{5} ( x )}{3}

解答ステップ

\frac{( 1 - cos ( x ) ) ^{3} cos ^{2} ( x )}{3}

(1 - cos (x))^{3} = 1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (x)

= \frac{cos ^{2} ( x ) ( 3 cos ^{2} ( x ) - cos ^{3} ( x ) - 3 cos ( x ) + 1 )}{3}

拡張

(1 - 3 cos (x) + 3 cos^{2} (x) - cos^{3} (x)) cos^{2} (x) : cos^{2} (x) - 3 cos^{3} (x) + 3 cos^{4} (x) - cos^{5} (x)

= \frac{cos ^{2} ( x ) - 3 cos ^{3} ( x ) + 3 cos ^{4} ( x ) - cos ^{5} ( x )}{3}

分数の規則を適用する:

\frac{a \pm b}{c} = \frac{a}{c} \pm \frac{b}{c}

\frac{cos ^{2} ( x ) - 3 cos ^{3} ( x ) + 3 cos ^{4} ( x ) - cos ^{5} ( x )}{3} = \frac{cos ^{2} ( x )}{3} - \frac{3 cos ^{3} ( x )}{3} + \frac{3 cos ^{4} ( x )}{3} - \frac{cos ^{5} ( x )}{3}

= \frac{cos ^{2} ( x )}{3} - \frac{3 cos ^{3} ( x )}{3} + \frac{3 cos ^{4} ( x )}{3} - \frac{cos ^{5} ( x )}{3}

数を割る:

\frac{3}{3} = 1

= \frac{cos ^{2} ( x )}{3} - cos^{3} (x) + cos^{4} (x) - \frac{cos ^{5} ( x )}{3}