Exponential Polar Form?

This qu involves writing cos(^3)x (not [cosx]^3 ... as I believe there's a difference?) in a different way using the Binomial Theorem. (x = theta)

I understand the process upto (1/8)*(e^3ix + 3e^ix + 3e^-ix + e^-3ix)

Then it states (1/4)*[0.5(e^3ix + e^-3ix) + 1.5(e^ix + e^-ix)]
which I derived my own way (they skipped some steps)..but I don't get the next part.

The next part states: (1/4)*(cos3x + 3cosx)

How?!

Thanks!

This qu involves writing cos(^3)x (not [cosx]^3 ... as I believe there's a difference?)