sinA - cosA +1 / sinA + cosA -1 = secA + tanA
Now secA = 1/cosA and tanA = sinA/cosA
So sinA - cosA +1 / sinA + cosA -1 = 1/cosA + sinA / cosA
From now on I'll write sinA = s and cosA = c :-
(s - c + 1 )/ (s + c - 1) = 1/c + s/c
(s - c + 1) / (s + c - 1) = (1 + s) / c
Cross multiply:-
s + c - 1 + s^2 + sc - s = sc - c^2 + c
s^2 + c + sc - 1 = sc - c^2 + c
s^2 - 1 + sc - sc + c - c = -c^2
s^2 - 1 = -c^2
-(1 - s^2) = - c^2
Now 1 - s^2 = c^2 so:-
- c^2 = - c^2
So the identity is proved