簡約 (e^{-i2x}-e^{-i4x})(e^{i2x}-e^{i4x})

解

簡約

(e^{- i 2 x} - e^{- i 4 x}) (e^{i 2 x} - e^{i 4 x})

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

解答ステップ

(e^{- i 2 x} - e^{- i 4 x}) (e^{i 2 x} - e^{i 4 x})

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

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

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

虚数の規則を適用する:

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

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

虚数の規則を適用する:

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

= (cos (2 x) - i sin (2 x) - (cos (4 x) - i sin (4 x))) (cos (2 x) + i sin (2 x) - (cos (4 x) + i sin (4 x)))

複素数の算術規則を適用する:

(a + bi) (a - bi) = a^{2} + b^{2}

a = cos (2 x) - cos (4 x), b = - sin (2 x) + sin (4 x)

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

簡素化

(cos (2 x) - cos (4 x))^{2} + (- sin (2 x) + sin (4 x))^{2} : (cos (2 x) - cos (4 x))^{2} + 4 cos^{2} (3 x) sin^{2} (x)

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