Question

Prove that (cos x + i sin x)^2 = cos2x + i sin2x

Answer 1

The trigonometric function is (cos x + i sin x)².

Expand the function.

(cos x + i sin x)² = (cos x)² + (i sin x)² +2i sinx cosx

= cos² x + i² sin² x + 2i sinx cosx

= cos² x -  sin² x + 2i sinx cosx                (Since i² = -1)

= (cos² x -  sin² x) + i (2 sinx cosx)

= cos 2x + i sin 2x                              (Trigonometric identities Cos 2x = cos² x -  sin² x and Sin 2x = 2 sinx cosx)

Therefore it is proved that (cos x + i sin x)² = cos 2x + i sin 2x.