Complex variables conformal mapping trig identity

[EnginerdRuns]

Homework Statement

map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation}
on some domain which contains z=e^{i\theta}. \theta between 0 and \pi
Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The resulting expression reduces to a simple trig function.

Homework Equations

I can get the map if I can figure out what function they're going for, but I have no idea what function this is.

The Attempt at a Solution

$$w = \Big(\frac{e^{i\theta}-1}{e^{i\theta}+1}\Big)^{2}$$
Where the heck do I go from here?

 

[tiny-tim]

Welcome to PF!

Hi EnginerdRuns! Welcome to PF!

Multiply top and bottom of the fraction by e^-iθ/2

 

[EnginerdRuns]

Thanks bro. I'm running off of way too little sleep at this point.