Physics: Probability density of interferometer

Hey. Problem here that I've spent hours trying to solve. I'm pretty sure that I understand the physics. It's just the trig in the maths that's tripping me up. I'm dissecting the question to provide the relevant info here.

http://img208.imageshack.us/img208/1...tquestion2.jpg

I've tried many things converting to trig functions etc (I did that a bit to go from the second last to last line)

Please can you check if my current working is correct and steer me in the right direction. I've tried so many different trig identities by hand and with scientific notebook for help and still....nothing.

thank you kindly in advance.