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Question

Prove that: 2cosπ13cos9π13+cos3π13+cos5π13=0

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Solution

L.H.S.

2cosπ13cos9π13+cos3π13+cos5π13

Using that,

2cosAcosB=cos(A+B)+cos(AB)

2cosπ13cos9π13+cos3π13+cos5π13

=cos(π13+9π13)+cos(π139π13)+cos3π13+cos5π13

=cos10π13+cos(8π13)+cos3π13+cos5π13cos(θ)=cosθ

=cos10π13+cos8π13+cos3π13+cos5π13

=cos(π3π13)+cos(π5π13)+cos3π13+cos5π13

=cos3π13cos5π13+cos3π13+cos5π13

=0

R.H.S
Hence, proved.

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