This one pops in mind, but it requires a cosine on the right end. Maybe it's a typo?
Or it could be that the phase is irrelevant, but the equals sign shouldn't be there anyway.
1) Always, always, ALWAYS ask if the answer makes sense!
Do the two expressions even agree with each other at the easily evaluated point of t=0?
No!
So you KNOW it is wrong! No point going any further. It's WRONG!
2) Had it passed the sanity test (may or may not be right, but at least it would have a fighting change), then you start looking for identities that involve the squares of the trig functions one angle and the first order trig functions of an angle twice as large. In other words, either the "double-angle" formulas or the "half-angle" formulas.
But since you already know that you CAN"T find an identity that will make this work out, there's no point in trying. What you CAN do is try to write the expression involving squared trig functions in terms of linear terms of the trig functions at twice the frequency, since that appears to be the intent.