What is factor e^{ix}-e^{-ix} ?

The solution to factor e^{ix}-e^{-ix} is (cos(-x)+isin(-x))(cos(x)+isin(x)+1)(cos(x)+isin(x)-1)

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factor e^{ix}-e^{-ix}

factor

e^{i x} - e^{- i x}

(cos (- x) + i sin (- x)) (cos (x) + i sin (x) + 1) (cos (x) + i sin (x) - 1)

Solution steps

e^{i x} - e^{- i x}

Apply exponent rule:

a^{b + c} = a^{b} a^{c}

e^{i x} = e^{- i x} e^{2 i x}

= e^{- i x} e^{2 i x} - e^{- i x}

Factor out common term

e^{- i x}

= e^{- i x} (e^{2 i x} - 1)

Factor

e^{2 i x} - 1 : (cos (x) + i sin (x) + 1) (cos (x) + i sin (x) - 1)

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

Refine

= (cos (- x) + i sin (- x)) (cos (x) + i sin (x) + 1) (cos (x) + i sin (x) - 1)

x + 2 = 4 - x^{2}

n^{2} + 20 n + 100

x^{2} + 4 x - 21 = 0

2 + \frac{5}{x} > 1

- x + 9 x = 32

What is factor e^{ix}-e^{-ix} ?
The solution to factor e^{ix}-e^{-ix} is (cos(-x)+isin(-x))(cos(x)+isin(x)+1)(cos(x)+isin(x)-1)