LHS = \(\frac{1+cos(A-B)\,cos\,C}{1+cos(A-C)\,cos\,B}\)
\(=\frac{1-cos(A-B)\,cos\,(A+B)}{1-cos(A-C)\,cos\,(A+C)}\)
\(=\frac{1-(cos^2\,A-sin^2\,B)}{1-(cos^2A-sin^2\,C)}\)
\(=\frac{sin^2\,B+1-cos^2\,A}{sin^2\,C+1-cos^2\,A}\)
= \(\frac{a^2+b^2}{a^2+c^2}\) = RHS
∴ LHS = RHS