Prove : (sinA-cosA+1)/(sinA+cosA-1) = (1+sinA)/cosA

multiply top and bottom of the left side by

(sinA + cosA + 1) and expand carefully

remember sin^2A + cos^2A = 1
and in the bottom replace the
-cos^2A with (1-sin^2A)

To prove the given equation:

(sinA - cosA + 1) / (sinA + cosA - 1) = (1 + sinA) / cosA

We need to manipulate the left side of the equation to match the right side.

Step 1: Simplify both the numerator and denominator.

Numerator:
sinA - cosA + 1

Denominator:
sinA + cosA - 1

Step 2: Multiply both the numerator and denominator by the conjugate of the denominator, which is (sinA + cosA + 1).

Numerator:
(sinA - cosA + 1) * (sinA + cosA + 1) = sin^2A + cos^2A + 1

Denominator:
(sinA + cosA - 1) * (sinA + cosA + 1) = sin^2A + cos^2A

Step 3: Simplify the numerator and denominator.

Numerator:
sin^2A + cos^2A + 1 = 1 + (sin^2A + cos^2A) = 1 + 1 = 2

Denominator:
sin^2A + cos^2A

Step 4: Rewrite the equation.

2 / (sin^2A + cos^2A) = (1 + sinA) / cosA

Step 5: Recall the identity sin^2A + cos^2A = 1.

2 / 1 = (1 + sinA) / cosA

Step 6: Simplify further.

2 = (1 + sinA) / cosA

Therefore, the given equation is proven.