[JEE Advanced 1978] express 1 1 cos theta 2i sin theta in the form x iy

Answer: $x=\dfrac1{5+3\cos\theta},\quad y=\dfrac{-2\sin\theta}{(1-\cos\theta)(5+3\cos\theta)}$.

Multiply by the conjugate: $|\text{den}|^2=(1-\cos\theta)^2+4\sin^2\theta=(1-\cos\theta)(5+3\cos\theta)$. Then $x=\frac{1-\cos\theta}{(1-\cos\theta)(5+3\cos\theta)}=\frac1{5+3\cos\theta}$ and $y=\frac{-2\sin\theta}{(1-\cos\theta)(5+3\cos\theta)}$.

JEE Advanced 1978 · Complex Numbers — verified solution by the Vidaara Team.