Finding the conjugate of an expression involving the reciprocal

Question

How do you find the conjugate of the following expression:

Here is my working, but I got bogged down by the algebra and think there must be an easier way.

Answer 1

There is no need to get a common denominator - leave the first term as $x+iy$. Then for the second term, do as you did and multiply by $\frac{x-iy}{x-iy}$ to get it in the form $a+ib$.

Now you have two terms bein added of the form $(x+iy)+(a+ib)$, which you can re-write as $(x+a)+i(y+b)$. Now you can take the conjugate.

Answer 2

It is $$x+iy+\frac{1}{x+iy}=\frac{((x+iy)^2+1)(x-iy)}{x^2+y^2}$$