Question

If `A+B+C=pi`, prove that `sin^2A+sin^2B+sin^2C= 2(1 +cos A cos B cosC)`

Answer 1

We have
`LHS=sin^(2)A+sin^(2)B+ sin ^(2)C`
` =(1)/(2)(1-cos 2A)+(1)/(2)(1-cos 2B)+(1)/(2)(1-cos 2C)`
`=(3)/(2)-(1)/(2)(cos2A+cos 2B+ cos 2C)`
`=(3)/(2)-(1)/(2)(-1-4cos A cos B cos C) ["See Example 3"]`
`=2+2cos A cosB cosC=2(1+cosA cosB cos C)=RHS.`
`sin^(2)A+ sin^(2)B+sin^(2)C=2(1+cosA cos B cos C).`