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Question

Prove cos3θ+cos3(120+θ)+cos3(220+θ)=34cos3θ

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Solution

We know that cos3A=4cos3A3cosA
cos3A=14(cos3A+3cosA).
Also cos(2nπ+θ)=cosθ.
Applying the above we have
L.H.S. =14[3cosθ+cos3θ)+3cos(120+θ)+cos(360+3θ)+3cos(240+θ)+cos(720+3θ)]
=14[3cos3θ]+34[cosθ+cos(120+θ)+cos(240+θ)]
=34cos3θ+34[cosθ+2cos(180+θ)cos60]
=34cos3θ+34[cosθ+2.12(cosθ)]
=34cos3θ.

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