Let z = cos ⁡ x + i sin ⁡ x. Show that 1 +...

Patatiniuh

Patatiniuh

Answered

2022-07-01

Let z = cos x + i sin x. Show that
1 + z = 2 cos 1 2 x ( cos 1 2 x + i sin 1 2 x )

Answer & Explanation

Miguidi4y

Miguidi4y

Expert

2022-07-02Added 13 answers

Since
cos x = 2 cos 2 x 2 1 sin x = 2 sin x 2 cos x 2 ,
we have
1 + z = 1 + cos x + i sin x = 1 + ( 2 cos 2 x 2 1 ) + i ( 2 sin x 2 cos x 2 ) = 2 cos 2 x 2 + 2 i sin x 2 cos x 2 = 2 cos x 2 ( cos x 2 + i sin x 2 )

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