Does the mod square of the wave function always have to be real?

[Plaetean]

Homework Statement

Doing a bit of QM from Griffiths intro to QM and got stuck early on on the following worked example:

http://imgur.com/6aPVGIr

I was under the impression that the mod square of the wave function ψ(x,t) should always be a positive, real number, but I cannot understand where the isin(2πt/h(E₂-E₁)) has gone in the last line. Perhaps I'm missing some simplification using trig identities perhaps?

Homework Equations

The Attempt at a Solution

 

[TSny]

In the next to last equality of the example, there is a product of two expressions in parentheses. Show us what you get when you multiply this out.