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If A+B+C=pi , prove that cos 2A +cos ...

If `A+B+C=pi `, prove that
`cos 2A +cos 2B +cos 2C=-1-4cos A cos Bcos C.`

Text Solution

Verified by Experts

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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