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

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

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expand (1-cos(4x))^4

expand

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

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

Solution steps

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

Apply binomial theorem:

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

a = 1, b = - cos (4 x)

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

(1 - cos (4 x))^{2}

(cos (θ) + i sin (θ))^{3}

\frac{\partial}{\partial x} ((f (x, y)) (x^{2} + y^{2}))

sin (x) (1 + tan (x))

(a + b + c + d + e)^{3}

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