What is (e^{16ix}-1)(e^{-8ix}+3e^{8ix}+e^{16ix}+3) ?

The solution to (e^{16ix}-1)(e^{-8ix}+3e^{8ix}+e^{16ix}+3) is

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simplify (e^{16ix}-1)(e^{-8ix}+3e^{8ix}+e^{16ix}+3)

simplify

(e^{16 i x} - 1) (e^{- 8 i x} + 3 e^{8 i x} + e^{16 i x} + 3)

(4 cos (16 x) cos (8 x) + cos^{2} (16 x) + 2 cos (16 x) - 4 cos (8 x) - 3 - 2 sin (16 x) sin (8 x) - sin^{2} (16 x)) + i (sin (32 x) + 2 sin (16 x) - 2 sin (8 x) + 2 cos (16 x) sin (8 x) + 4 cos (8 x) sin (16 x))

Solution steps

(e^{16 i x} - 1) (e^{- 8 i x} + 3 e^{8 i x} + e^{16 i x} + 3)

Apply imaginary number rule:

e^{ia} = cos (a) + i sin (a)

= (cos (16 x) + i sin (16 x) - 1) (e^{- 8 i x} + 3 e^{8 i x} + e^{16 i x} + 3)

e^{- 8 i x} = cos (8 x) - i sin (8 x)

3 e^{8 i x} = 3 (cos (8 x) + i sin (8 x))

e^{16 i x} = cos (16 x) + i sin (16 x)

= (cos (16 x) + i sin (16 x) - 1) (cos (8 x) + cos (16 x) + i sin (16 x) + 3 (cos (8 x) + i sin (8 x)) - i sin (8 x) + 3)

Apply complex arithmetic rule:

(a + bi) (c + d i) = (a c - b d) + (a d + b c) i

a = cos (16 x) - 1, b = sin (16 x), c = 4 cos (8 x) + cos (16 x) + 3, d = 2 sin (8 x) + sin (16 x)

= ((cos (16 x) - 1) (4 cos (8 x) + cos (16 x) + 3) - sin (16 x) (2 sin (8 x) + sin (16 x))) + ((cos (16 x) - 1) (2 sin (8 x) + sin (16 x)) + sin (16 x) (4 cos (8 x) + cos (16 x) + 3)) i

Simplify

(cos (16 x) - 1) (2 sin (8 x) + sin (16 x)) + sin (16 x) (4 cos (8 x) + cos (16 x) + 3) : 2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)

= ((cos (16 x) - 1) (4 cos (8 x) + cos (16 x) + 3) - sin (16 x) (2 sin (8 x) + sin (16 x))) + (2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)) i

Remove parentheses:

(a) = a

= (cos (16 x) - 1) (4 cos (8 x) + cos (16 x) + 3) - sin (16 x) (2 sin (8 x) + sin (16 x)) + (2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)) i

Rewrite

(cos (16 x) - 1) (4 cos (8 x) + cos (16 x) + 3) - sin (16 x) (2 sin (8 x) + sin (16 x)) + (2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)) i

in standard complex form:

(4 cos (16 x) cos (8 x) + cos^{2} (16 x) + 2 cos (16 x) - 4 cos (8 x) - 3 - 2 sin (16 x) sin (8 x) - sin^{2} (16 x)) + (2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)) i

= (4 cos (16 x) cos (8 x) + cos^{2} (16 x) + 2 cos (16 x) - 4 cos (8 x) - 3 - 2 sin (16 x) sin (8 x) - sin^{2} (16 x)) + (2 cos (16 x) sin (8 x) + sin (32 x) - 2 sin (8 x) + 2 sin (16 x) + 4 sin (16 x) cos (8 x)) i

- \frac{( 6 x - 5 ) ( x + 5 )}{( x - 6 ) ( x + 6 ) ( x - 5 )}

2 (5 - 3)

\frac{3 b ( a ^{2} + 5 )}{c}

2 (33 - 67 + 21 - 14)

x l e f t (x + y)

What is (e^{16ix}-1)(e^{-8ix}+3e^{8ix}+e^{16ix}+3) ?
The solution to (e^{16ix}-1)(e^{-8ix}+3e^{8ix}+e^{16ix}+3) is