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Question

The value of 2 cosπ13 cos 9π13+cos 3π13+cos 5π13 is ________________.

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Solution

2 cosπ13 cos 9π13+cos 3π13+cos 5π13
Using identity:- 2 cos x cos y = cos (x + y) + cos (x – y)
=cos9π13+π13+cos9π13-π13+cos3π13+cos5π13=cos10π13+cos8π13+cos3π13+cos5π13=cos10π13+cos3π13+cos8π13+cos5π13using identity, cos x+cos y=2cosx+y2 cosx-y2=2cos10π13+3π132 cos10π13-3π132+2cos8π13+5π132 cos8π13-5π132=2cos13π132 cos7π132+2cos13π132 cos4π132=2cos π2 cos 7π26+2cosπ2 cos4π26=2cosπ2cos7π26+4π26=2×0cos7π26+cos4π26=0
Hence, the answer is 0.

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