Electrodynamics Fourier Analysis (Fouriers Trick)

[Tphysics]

Homework Statement

Two infinitely grounded metal plates at y=0 and y=a are connected at x=b and x=-b by metal strips maintained at a constant potential V. Find the potential inside the rectangular pipe.

Homework Equations

Laplaces Equation

The Attempt at a Solution

I posted a photo of what I've done so far. The line indicates where I stray off I'm confused because how can I use fouriers trick with cosh?

 

[Tphysics]

This is also example 3.4 in Griffiths E&M. He just breezes over the Fourier analysis part saying its the same as example 3.3 which is in this photo. However my cosh term doesn't cancel so I'm not just left with the sine term?