Prove that: `sin6^0sin42^0sin 66^0sin 78^0=1/(16)`

Explanation

L.H.S. `=frac{1}{2}[(2 sin { }^{circ} sin 66^{circ})(2 sin 42^{circ} sin 78^{circ})]`


`=frac{1}{4}[(cos 60^{circ}-cos 72^{circ})(cos 36^{circ}-cos 120^{circ})]`


`=frac{1}{4}[(frac{1}{2}-frac{sqrt{5}-1}{4})(frac{sqrt{5}+1}{4} frac{1}{2})]`


`=frac{1}{4} frac{3-sqrt{5}}{4} cdot frac{3+sqrt{5}}{4}=frac{1}{64}(9-5)=frac{1}{16}`