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Question

If cos2π3cos(π)+cos4π3cos5π3+cos(2π)cos7π3++cos40π3cos41π3=k, then the value of |6k| is

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Solution

k=cos2π3cos(π)+cos4π3cos5π3+cos(2π)cos7π3++cos40π3cos41π3
=cos2π3+cos4π3+cos(2π)++cos40π3 (cos(π)+cos5π3+cos7π3++cos41π3)
=19n=0cos(2π3+cos2nπ3)19n=0cos(π+cos2nπ3)
=sin(80π3)cos(2π3+19π3)sin(π3)sin(80π3)cos(π+19π3)sin(π3)
=sin(π3)cos(7π)sin(π3)sin(π3)cos(2π3)sin(π3)
=1(12)
=12
|6k|=3

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