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Question

If log254k=1log2(sinkπ5)=pq, where p and q are co-prime, then the value of p+q is

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Solution

E=log254k=1log2(sinkπ5)=log25(log2sinπ5+log2sin2π5+log2sin3π5+log2sin4π5)=log25(log2(sin2π5sin22π5))=log25log2(14(2sin2π52sin22π5))=log25log2(14(1cos72)(1cos144))=log25log2(14(1514)(1+5+14))=log25log2(14(54))=log25log2(516)=log25log25+4E=41=pq

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