Question

If `A+B+C=180^0`, prove that : `cos^2 A+cos^2 B-cos^2 C=1-2sinA sinB cosC`

Answer 1

We have
`LHS=cos^(2)A+cos^(2)B-cos^(2)C`
`=(1)/(2)(1+cos2A)+(1)/(2)(1+cos2B)-(1)/(2)(1+cos2C)`
`=(1)/(2)+(1)/(2)(cos2A+cos2B-cos2C)`
`=(1)/(2)+(1)/(2)(1-4sin A sin B cos C)["See Example4"]`
`=1-2 sin A sin B cosC=RHS.`
`therefore cos^(2)A+ cos^(2)B-cosC=1-2sinA sin B cosC.`