Question

proove that 1/1-tanA+1/1-cosA=1

Answer 1

Given expression on LHS is (tanA+secA-1) / (tanA-secA+1) = (1+sinA) / cosA.
Multiply and divide the above by (tanA+secA + 1) / (tanA+secA +1).
Using the algebraic identity (a+b)(a-b)=(a^2 - b^2)
(tanA+secA-1) / (tanA-secA+1) × (tanA+secA + 1) / (tanA+secA +1).=
[(tanA+secA)^2-1^2] / [(1+tanA)^2-sec^2 A] =
[tan^2 A + sec^2 A + 2 tanASecA - 1] / [1 + tan^2 A + 2tanA - sec^2 A] =
[tan^2 A + 1 + tan^2 A + 2 tanASecA - 1] / [1 + tan^2 A + 2tanA -sec^2 A] =
2tanA (tanA + secA) / 2 tan A = (tanA + secA) = (sinA/cosA + 1/cosA) = (1+sinA)/cosA = R H S  proved