If a = cosθ + isinθ, find the value of `(1 + "a")/(1 - "a")`.
Solution
Given that: a = cosθ + isinθ
∴ `(1 + a)/(1 - a) = (1 + cos theta + i sin theta)/(1 - cos theta - i sin theta)`
= `(1 + cos theta + i sin theta)/(1 - cos theta - i sin theta) xx (1 - cos theta + i sin theta)/(1 - cos theta + i sin theta)`
= `(1 - cos theta + i sin theta + cos theta - cos^2 theta + i sin theta cos theta + i sin theta - i sin theta cos theta + i^2 sin^2 theta)/((1 - cos theta)^2 - i^2 sin^2 theta)`
= `(1 + i sin theta - cos^2 theta + i sin theta - sin^2 theta)/(1 + cos^2 theta - 2 cos theta + sin^2 theta)`
= `(sin^2 theta + 2i sin theta - sin^2 theta)/(1 + 1 - 2 cos theta)`
= `(2i sin theta)/(2 - 2 cos theta)`
= `(2i sin theta)/(2(1 - cos theta))`
= `(i sin theta)/(1 - cos theta)`
= `(2 sin theta/2 cos theta/2.i)/(2sin^2 theta/2)`
= `cot theta/2 . i`
Hence, `(1 + a)/(1 - a) = icot theta/2`.
