How do we get double angle identities?

Question

how are the double angle identitites derived from the addition identities?  I am trying to find the formula for for sin3A. If Sin2A=2SinACosA is Sin3A just Sin3ACosA??

Answer 1

sin(3A)=sin(2A+A)=sin(2A)cos(A)+cos(2A)sin(A)=

2sin(A)cos²(A)+(1-2sin²(A))sin(A)=

2sin(A)(1-sin²(A))+sin(A)-2sin³(A)=

2sin(A)-2sin³(A)+sin(A)-2sin³(A)=

3sin(A)-4sin³(A).

SPOT CHECK

If A=π/6, 3A=π/2.

sin(3A)=1, sin(A)=½.

3sin(A)-4sin³(A)=3/2-4/8=3/2-1/2=1=sin(π/2)=sin(3A).