4sinαsin(π/3+α)sin(π/3-α)=sin3α

[latex]4sin alpha*sin(frac{pi}{3}+alpha)*sin(frac{pi}{3}-alpha)=\\ 4sin alpha*frac{cos((frac{pi}{3}+alpha)-(frac{pi}{3}-alpha))-cos((frac{pi}{3}+alpha)+(frac{pi}{3}-alpha))}{2}}=\\ 2sin alpha*(cos(frac{pi}{3}+alpha-frac{pi}{3}+alpha)-cos(frac{pi}{3}+alpha+frac{pi}{3}-alpha))=\\ 2sin alpha*(cos(2alpha)-cos(frac{2pi}{3}))=\\ 2sin alpha*(cos(2alpha)-cos(pi-frac{pi}{3}))=\\ [/latex] [latex]2sin alpha *(cos(2alpha)+cos(frac{pi}{3}))=\\ 2sin alpha *(1-2sin^2 alpha+frac{1}{2})=\\ sin alpha *(2-4sin^2 alpha+1)=\\ sin alpha *(3-4sin^2 alpha)=\\ 3sin alpha-4sin^3 alpha=sin (3alpha)[/latex]