簡約-(e^{ix}+e^{-ix})i(e^{ix}-e^{-ix})

解

簡約

- (e^{i x} + e^{- i x}) i (e^{i x} - e^{- i x})

4 cos (x) sin (x)

解答ステップ

- (e^{i x} + e^{- i x}) i (e^{i x} - e^{- i x})

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

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

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

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

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

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

簡素化

cos (x) + sin (x) i + cos (x) - i sin (x) : 2 cos (x)

= - 2 i cos (x) (cos (x) + i sin (x) - (cos (x) - i sin (x)))

拡張

cos (x) + sin (x) i - (cos (x) - i sin (x)) : 2 i sin (x)

= - 2 \cdot 2 ii cos (x) sin (x)

数を乗じる:

2 \cdot 2 = 4

= - 4 ii cos (x) sin (x)

指数の規則を適用する:

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

ii = i^{1 + 1}

= - 4 cos (x) i^{1 + 1} sin (x)

数を足す:

1 + 1 = 2

= - 4 cos (x) i^{2} sin (x)

i^{2} = - 1

= - 4 (- 1) cos (x) sin (x)

改良

= 4 cos (x) sin (x)