given that

(1+cosA)(1+cosB)(1+cosC) = (1-cosA)(1-cosB)(1-cosC)

Dividing both sides with (1-cosa)(1-cosb)(1-cosc), we get:

 

 

(1+cosa)(1+cosb)(1+cosc) / (1-cosa)(1-cosb)(1-cosc) = 1

 

Multiplying the numerator and the denominator by (1-cosa)(1-cosb)(1-cosc), we get:

 

{(1+cosa)(1-cosa)(1+cosb)(1-cosb)(1+cosc)(1-cosc)} / {(1-cosa)(1-cosa)(1-cosb)(1-cosb)(1-cosc)(1-cosc)}

 

Using the identity sin^2 x + coa^2 x = 1 and taking square root on both sides, we get

(sina.sinb.sinc) = (1-cosa)(1-cosb)(1-cosc)

 

Thus, we have:

(1+cosA)(1+cosB)(1+cosC) = (1-cosA)(1-cosB)(1-cosC) = sina sinb sinc

 

Hence the result.