Prove that: `sin(pi/5)sin(2pi/5)sin(3pi/5)sin(4pi/5)=5/16`

Explanation

`L.H.S` `=sin frac{pi}{5} cdot sin frac{2 pi}{5} cdot sin frac{3 pi}{5} cdot sin frac{4 pi}{5}`


`=sin 36^{circ} cdot sin 72^{circ} cdot sin 108^{circ} cdot sin 144^{circ}`


`=sin 36^{circ} cdot sin(90^{circ}-18^{circ}) cdot sin(90^{circ}+18^{circ}) cdot sin(180^{circ}-36^{circ})`


`=sin 36^{circ} cdot cos 18^{circ} cdot cos 18^{circ} cdot sin 36^{circ}`


`=sin^{2} 36^{circ} cdot cos^{2} 18^{circ}`


`=[frac{sqrt{10-2 sqrt{5}}}{4}]^{2}[frac{sqrt{10+2 sqrt{5}}}{4}]^{2}`


`=frac{(10-2 sqrt{5})}{16} cdot frac{(10+2 sqrt{5})^{2}}{16}`


`=frac{100-(2 sqrt{5})^{2}}{16.16}=frac{100-20}{16.16}`


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