(a² + b²) / (a² + c²) = [(1 + cos(A − B)) × cos(C)] / [(1 + cos(A − C)) × cos(B)]

L.H.S. = (a² + b²) / (a² + c²)

= (k² × sin²A + k² × sin²B) / (k² × sin²A + k² × sin²C)

= (sin²A + sin²B) / (sin²A + sin²C)

= [(1 − cos(2A))/2 + (1 − cos(2B))/2] / [(1 − cos(2A))/2 + (1 − cos(2C))/2]

= [2 − (cos(2A) + cos(2B))] / [2 − (cos(2A) + cos(2C))]

= [2 − 2·cos(A + B) × cos(A − B)] / [2 − 2·cos(A + C) × cos(A − C)]

= [1−cos(π−C) × cos(A−B)] / [1−cos(π−B) × cos(A−C)] (since A+B+C=π)

= (1 + cos(A − B) × cosC) / (1 + cos(A − C) × cosB) = R.H.S.