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If `A+B+C=pi `, prove that `cos 2A +cos 2B +cos 2C=-1-4cos A cos Bcos C.` |
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Answer» We have `LHS=cos 2A+cos 2B+cos 2C` `=2cos(A+B)cos (A-B)+2cos^(2)C-1` `=2cos (pi-C)cos (A-B)+2cos^(2)C-1` `=-2cos C cos (A-B)+2cos^(2)C-1` `=-2cos C[cos (A-B)-cosC]-1` `=-1-2cos C[cos (A-B)-cos {pi-(A-B)}]` `=-1-2cos C[cos(A-B)+cos(A+B) ]` `=-1-4cos A cos B cos C =RHS.` `therefore cos 2A +cos 2B +cos2C=-1-4cos A cos B cos C.` |
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