Question

If `A+B+C=pi`, prove that : `cos2A+cos2B-cos2C=1-4sinA sinB cosC`

Answer 1

We have
`LHS=(cos2A+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`
`=1-2 cos C [ cos (A-B)+cosC]`
`=1-2cos C [cos(A-B)+cos{pi-(A+b)}]`
`=1-2cos C [ cos (A-B)-cos (A+B)]`
`=1-2 cos C [2sin A sin C=RHS.`
`therefore cos 2A+cos2B-cos2C=1-4sin A sin B cos C.`