Prove That,

cosπ7cos2π7+cos3π7=12

L.H.S.=cosπ7cos2π7+cos3π7

= cosπ7+cos3π7cos(π5π7)

= cosπ7+cos3π7+cos5π7

= cos(π7+312.2π7).sin(3.2π/72)sin(2π/72)

= cos(π7+2π7).sin3π7sinπ7

= cos3π7sin3π7sinπ7=2cos3π7sin3π72sinπ7

= sin6π72sinπ7=sin(ππ7)2sinπ7=12

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