I tend to forget some of the trigonometric functions and someone showed me how to derive the double angle identities from what I think is Euler's formula:
=
=
I have a question about this step...I understand how the '2' from the e^(i2x) was pulled out, but is it okay to pull the '2' out from the '2x' of the trigonometric functions and to square the right hand expression? How does that work?
In fact, what is written in OP does not require DeMoivre's Theorem. From Euler's formula, we have
(1)
and
(2)
Squaring both sides of (1), we get
(3)
Now we compare the right-hand side of (2) and (3) and equate the real and imaginary parts to derive the identities for double-angle sine and cosine.