What is expand (1+cos(x))^4 ?

The solution to expand (1+cos(x))^4 is 1+4cos(x)+6cos^2(x)+4cos^3(x)+cos^4(x)

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expand (1+cos(x))^4

expand

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

1 + 4 cos (x) + 6 cos^{2} (x) + 4 cos^{3} (x) + cos^{4} (x)

Solution steps

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

Apply binomial theorem:

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

a = 1, b = cos (x)

= i = 0 \sum 4 (i 4) \cdot 1^{(4 - i)} cos^{i} (x)

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} \cdot 1^{4} cos^{0} (x) : 1

Simplify

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

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} \cdot 1^{2} cos^{2} (x) : 6 cos^{2} (x)

Simplify

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

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} \cdot 1^{0} cos^{4} (x) : cos^{4} (x)

= 1 + 4 cos (x) + 6 cos^{2} (x) + 4 cos^{3} (x) + cos^{4} (x)

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

(x^{2} + 2 x) e^{x}

3 (22 - 1)

(1 + 2 sin (x))^{5}

(1 + 2 sin (x))^{2}

What is expand (1+cos(x))^4 ?
The solution to expand (1+cos(x))^4 is 1+4cos(x)+6cos^2(x)+4cos^3(x)+cos^4(x)