This isn't an identity, so I assume you have to solve the equation.
(1 - sin(2A)) (1 + cot(2A)) = cot(2A)
1 - sin(2A) + cot(2A) - sin(2A) cot(2A) = cot(2A)
1 - sin(2A) - cos(2A) = 0
sin(2A) + cos(2A) = 1
Multiply both sides by 1/√2, which we want to do because cos(π/4) = sin(π/4) = 1/√2. This gives
cos(π/4) sin(2A) + sin(π/4) cos(2A) = 1/√2
Then condense the left side as
sin(2A + π/4) = 1/√2
2A + π/4 = sin⁻¹(1/√2) + 2nπ or 2A + π/4 = π - sin⁻¹(1/√2) + 2nπ
(where n is any integer)
2A + π/4 = π/4 + 2nπ or 2A + π/4 = 3π/4 + 2nπ
2A = 2nπ or 2A = π/2 + 2nπ
A = nπ or A = π/4 + nπ
