Question

Prove that
(tanA-tanB)² + (1+tanAtanB)^{2
is equal to sec square A*sec squareB}

Answer 1

Taking L.H.S. , we have
LHS = (1 + tan a tan b)2 + (tan a – tan b)2
= 1 + tan2a tan2b + 2 tan a tan b + tan2a + tan2b – 2 tan a tan b
= 1 + tan2a tan2b + tan2a + tan2b
= 1. (1 + tan2b) + tan2a (tan2b + 1)
= (1 + tan2b) + (1 + tan2a)
= sec2b sec2a
= sec2a sec2b
= RHS