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Question

Find : cos2π7+cos4π7+cos6π7+32

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Solution

cos2π7+cos4π7+cos6π7+12
to multiply by sinπ7
sinπ7cos2π7+sinπ7cos4π7+sinπ7cos6π7+12sinπ7
sinAcosB=12[sin(AB)+sin(A+B)]
12[sin(π22π7)+sin(π7+2π7)]
+12[sin(π74π7)+sin(π7+4π7)]
+12[sin(π76π7)+sin(π7+6π7)]+12sinπ7
multiply by 2:
sinπ7+sin3π7sin3π7+sin5π7sin5π7+sin7π7+sinπ7
sinπ7+sinπ+sinπ7=0 321=12
cos2π7+cos4π7+cos6π7+12=0
cos2π7+cos4π7+cos6π7+12+1=0+1=1.

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