Question

Derive an identity that transforms sin(α+β+γ) into a sum of products of sines and/or cosines of the individual numbers α, β, γ.

Answer 1

sin(α+β+γ) , I will use sin(a+b+c) for easier typing
= sin( (a+b) +c)
= sin(a+b)cos c + cos(a+b)sin c
= (sinacosb + cosasinb)(cosc) + (cosacosb - sinasinb)sinc
= sinacosbcosc + cosacoscsinb + cosacosbsinc - sinasinbsinc

now put back your α,β, and γ