Question

If a=cos theta+isin theta find the value of (1+a)/(1-a)

Answer 1

Answer:

icotθ/2

Step-by-step explanation:

a = cosθ + i sinθ

Now 1+a/1-a

=> 1+ cosθ + i sinθ / 1 - cosθ - i sinθ)

//multiply numerator and denominator by 1 - cosθ + i sinθ

=> (1+ cosθ + i sinθ)(1 - cosθ + i sinθ)/(1 - cosθ + i sinθ)(1 - cosθ - i sinθ)

=> [1 + cosθ + i sinθ - cosθ - cos²θ - isinθcosθ+isinθ+isinθcosθ+(isinθ)²] / [(1-cosθ)² - (isinθ)²]

=> [1+2isinθ-cos²θ-sin²θ] / {1-2cosθ+cos²θ-(i²sin²θ)}

=> (1+2isinθ- (cos²θ+sin²θ)] /  {1-2cosθ+cos²θ+ sin²θ]

=> (1+2isinθ-1)/(1-2cosθ+1)

=> 2isinθ/(2-2cosθ)

=> isinθ/(1-cosθ)

//remeber sin2A = 2sinAcosA

=> i(2sinθ/2cosθ/2)/(2sin²θ/2)

=> (icosθ/2)/(sinθ/2)

=> icotθ/2.