Prove that `4sinthetasin(pi/3+theta)sin((2pi)/3+theta)=sin3theta`

Explanation

`4sinthetasin (frac{pi}{3}+theta) sin (frac{pi}{3}-theta)`


`4 sinthetasin (frac{pi}{3}+theta) sin (pi-(frac{pi}{3}-theta))`


`4sinthetasin(frac{pi}{3}+theta) sin (frac{2 pi}{3}+theta)`


`=4 sin theta(sin theta cos frac{pi}{3}+cos theta sin frac{pi}{3})(sin theta cos frac{2 pi}{3}+cos theta sin frac{2 pi}{3})`


`=4 sin theta(frac{sin theta}{2}+frac{sqrt{3} cos theta}{2})(-frac{sin theta}{2}+frac{sqrt{3} cos theta}{2})`


`=4 sin theta(frac{3}{4} cos ^{2} theta-frac{1}{4} sin ^{2} theta)`


`=sin theta(3 cos ^{2} theta-sin ^{2} theta)`


`=sin theta(3(1-sin ^{2} theta)-sin ^{2} theta)`


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