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受欢迎的代数 >

展开 (cos(x)+sin(x))^4

解答

展开

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

解答

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

求解步骤

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

使用二项式定理:

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

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

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

展开求和

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

化简

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

化简

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

化简

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

化简

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

化简

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

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

流行的例子

展开 (x^{3/2}+1)^2

e x p an d (x^{\frac{3}{2}} + 1)^{2}

展开 (x^4+3x^2)/((x^2+1)^2)

e x p an d \frac{x ^{4} + 3 x ^{2}}{( x ^{2} + 1 ) ^{2}}

展开 sin(x)cos(x)(sec(x)+cot(x))

e x p an d sin (x) cos (x) (sec (x) + cot (x))

化简 x(1-4/(x+3))

s im pl i f y x (1 - \frac{4}{x + 3})

展开 (1+2/3 x)^6(3+6x)^2

e x p an d (1 + \frac{2}{3} x)^{6} (3 + 6 x)^{2}