In any triangle `A B C` , prove that: `(1+cos(A-B)cos C)/(1+cos(A-C)cos B)=(a^2+b^2)/(a^2+c^2)`

Explanation

Given,

`LHS = (1+cos(A−B)cosC)/(1+cos(A−C)cosB )`

=`(1−cos(A−B)cos(A+B))/(1−cos(A−C)cos(A+C) )`

=`(1−(cos^2A−sin^2B))/(1−(cos^2A−sin^2C) )`

=`(sin^2B+1−cos^2A)/(sin^2C+1−cos^2A)`

=`(sin^2B+sin^2A)/(sin^2C+sin^2A)...(i)`

let `sinA =ka,sin B =kb,sin C=kc` so,equation (i) becomes

= `(a^2+b^2)/(a^2+c^2) = RHS`