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Question

The value of cos2π6+θ- sin2π6-θ is
(a) 12 cos 2 θ

(b) 0

(c) -12 cos 2 θ

(d) 12

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Solution

(a) 12 cos 2 θ

We have,cos2π6+θ-sin2π6-θ=cos2π6+θ-cos2π2-π6-θ=cos2π6+θ-cos2π3+θ=cosπ6+θ+cosπ3+θcosπ6+θ-cosπ3+θ=2cosπ6+θ+π3+θ2 cosπ6+θ-π3-θ2 2sinπ6+θ+π3+θ2 sinπ3+θ-π6-θ2=4cosπ4+θcos-π12 sinπ4+θ sinπ12=4cosπ4+θcosπ12 sinπ4+θ sinπ12=2sinπ4+θcosπ4+θ2 sinπ12cosπ12=sinπ2+2θsinπ6=cos2θ×12=12cos2θ

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