The value oif the expressin `(1+cospi/10)(1+cos(3pi)/10)(1+cos(7pi)/10)(1+cos(9pi)/10)` is

Explanation

`(1-cos frac{pi}{10})(1+cos frac{3 pi}{10})(1+cos frac{7 pi}{10})(1-cos frac{9 pi}{10})`


`=(1-cos frac{pi}{10})(1+cos frac{3 pi}{10})(1-cos frac{3 pi}{10})(1+cos frac{pi}{10})`


`=(1-cos ^{2} frac{pi}{10})(1-cos ^{2} frac{3 pi}{10})`


`=sin ^{2} frac{pi}{10} sin ^{2} frac{3 pi}{10}`


`=(sin frac{pi}{10} sin frac{3 pi}{10})^{2}`


`=(sin 18^{circ} sin 54^{circ})^{2}`


`=(sin 18^{circ} cos 36^{circ})^{2}`


`=(frac{sqrt{5}-1}{4} cdot frac{sqrt{5}+1}{4})^{2}`


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