Let’s first deal with double-angle formulas, which break an expression involving a single angle into one involving two angles, and prove at least one of them. The proof is short and easy, which is a good starting point for this chapter.
Prove that sin2u = 2sinucosu.
To prove this identity, let v = u in the sum formulas for sines. Then obtain the following:
sin2u = sin(u + u) = sinucosu + cosusinu = 2sinucosu
Note that the order of the factors does not change the product; thus sinucosu = cosusinu.
You proved your first double-angle formula. The other double-angle formulas can be proved in a similar way. Let’s list all three of them.
The following are called double-angle formulas:
sin2u = 2sinucosu
cos2u = cos2u – sin2u = ...