# If a = cosθ + isinθ, find the value of aa1+a1-a. - Mathematics

Sum

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.

Concept: Concept of Complex Numbers
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#### APPEARS IN

NCERT Mathematics Exemplar Class 11
Chapter 5 Complex Numbers and Quadratic Equations
Exercise | Q 6 | Page 91

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