What is expand (cos(θ)+jsin(θ))^6 ?

The solution to expand (cos(θ)+jsin(θ))^6 is

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expand (cos(θ)+jsin(θ))^6

expand

(cos (θ) + j sin (θ))^{6}

cos^{6} (θ) + 6 j cos^{5} (θ) sin (θ) + 15 j^{2} cos^{4} (θ) sin^{2} (θ) + 20 j^{3} cos^{3} (θ) sin^{3} (θ) + 15 j^{4} sin^{4} (θ) cos^{2} (θ) + 6 j^{5} sin^{5} (θ) cos (θ) + j^{6} sin^{6} (θ)

Solution steps

(cos (θ) + j sin (θ))^{6}

Apply binomial theorem:

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

a = cos (θ), b = j sin (θ)

= i = 0 \sum 6 (i 6) cos^{(6 - i)} (θ) (j sin (θ))^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} cos^{6} (θ) (j sin (θ))^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} cos^{5} (θ) (j sin (θ))^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} cos^{4} (θ) (j sin (θ))^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} cos^{3} (θ) (j sin (θ))^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} cos^{2} (θ) (j sin (θ))^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} cos^{1} (θ) (j sin (θ))^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} cos^{0} (θ) (j sin (θ))^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} cos^{6} (θ) (j sin (θ))^{0} : cos^{6} (θ)

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} cos^{5} (θ) (j sin (θ))^{1} : 6 j cos^{5} (θ) sin (θ)

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} cos^{4} (θ) (j sin (θ))^{2} : 15 j^{2} cos^{4} (θ) sin^{2} (θ)

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} cos^{3} (θ) (j sin (θ))^{3} : 20 j^{3} cos^{3} (θ) sin^{3} (θ)

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} cos^{2} (θ) (j sin (θ))^{4} : 15 j^{4} sin^{4} (θ) cos^{2} (θ)

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} cos^{1} (θ) (j sin (θ))^{5} : 6 j^{5} sin^{5} (θ) cos (θ)

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} cos^{0} (θ) (j sin (θ))^{6} : j^{6} sin^{6} (θ)

= cos^{6} (θ) + 6 j cos^{5} (θ) sin (θ) + 15 j^{2} cos^{4} (θ) sin^{2} (θ) + 20 j^{3} cos^{3} (θ) sin^{3} (θ) + 15 j^{4} sin^{4} (θ) cos^{2} (θ) + 6 j^{5} sin^{5} (θ) cos (θ) + j^{6} sin^{6} (θ)

\frac{( sec ( x ) + sin ( x ) ) cot ( x )}{csc ( x ) + cos ( x )}

- \frac{( x + 1 ) ( x - 2 ) ( x - 5 )}{8}

3 x - 2 \cdot (3 x - 2)

(4 + 7) (8 - 7)

x^{2} + 3 x - 28 x^{2} + 3 x - 28

What is expand (cos(θ)+jsin(θ))^6 ?
The solution to expand (cos(θ)+jsin(θ))^6 is