Complex Number Question?

Question

This Question is really bugging me:

Let z = cos A + iSinA

Express 2/(1+z) in the form 1 - itan(kA), z not equal to -1.

Can anybody solve this please?

Answer 1

Put w = 1 + z = 1 + cosA + i sin A. The conjugate of w, w', is w' = 1 + cosA - i sin A and
|w|^2 = 1 +2cosA + (cosA^2 + (sinA)^2 = 2(1 + cosA).

So,

2/(1+z) = 2/w = 2w'/(|w|^2 )= 2(1 +cosA- isinA)/(2(1+cosA) = (1 + cosA -isinA)/(1 + cosA) = 1 - i sinA/(1 + cosA).

From Trigonometry,

sinA = 2sin(A/2) cos(A/2)
1 + cosA = 2(cosA)^2

Hence,

sinA/(1 + cosA) = (2sin(A/2) cos(A/2))/( 2(cosA)^2) = tan(A/2).

Finally,

2/(1+z) = 1 - i tan(A/2), which is the given expression with k =1/2