I'd make just a couple of small changes so that the solution agrees with the standard setting out of an identity proof:
And here's where we reach the fork in the road because you need to keep working with the R.H.S.
Do you see the style required?
In my opinion, trig identities should follow the format:
This side (LHS or RHS as required) = whatever it is
= blah blah
= yada yada
= ho hum ho hum
= the other side (RHS or LHS as required).