(sin A-cos A+1)/(sin A+cosA-1)=1/(sec A-tan A) L.H.S. divide above and below by cos A =(tan A-1+secA)/(tan A+1-sec A) =(tan A-1+secA)/(1-sec A+tan A) We know that 1+tan^2A=sec ^2A 1=sec^2A-tan ^2A=(sec A+tan A)(secA-tanA) =(sec A+tan A-1)/[(sec A+tan A)(sec A-tan A)-(sec A-tan A)] =(sec A+tan A-1)/(sec A-tan A)(sec A+tan A-1) = 1/(sec A-tan A) , proved.

Proof:

To prove the equation sin A - cos A / (1/sin A - cos A) = 1 / (sec A - tan A) using the identity sec^2A = 1 - tan^2A, we need to manipulate the left side of the equation and simplify it until it matches the right side.

Let's start by simplifying the left side of the equation:

Step 1: Rewrite the left side using the reciprocal identities:
sin A - cos A / (1/sin A - cos A) = sin A - cos A * (sin A / (1 - cos A * sin A))

Step 2: Simplify the expression:
= sin A - cos A * sin A / (1 - cos A * sin A)

Step 3: Combine the terms with sin A:
= (sin A - cos^2A * sin A) / (1 - cos A * sin A)

Step 4: Factor out sin A:
= sin A * (1 - cos^2A) / (1 - cos A * sin A)

Step 5: Apply the identity sin^2A + cos^2A = 1:
= sin A * sin^2A / (1 - cos A * sin A)

Step 6: Simplify the expression:
= (sin^3A) / (1 - cos A * sin A)

Now, let's manipulate the right side of the equation:

Step 7: Rewrite the right side using the reciprocal identities:
1 / (sec A - tan A) = 1 / (1/cos A - sin A / cos A)

Step 8: Simplify the expression:
= cos A / (1 - sin A)

Step 9: Multiply the numerator and denominator by cos A:
= cos A * cos A / (cos A - sin A * cos A)

Step 10: Apply the identity cos^2A = 1 - sin^2A:
= cos^2A / (cos A - sin A * cos A)

Step 11: Simplify the expression:
= (1 - sin^2A) / (cos A - sin A * cos A)

Step 12: Apply the identity 1 - sin^2A = cos^2A:
= cos^2A / (cos A - sin A * cos A)

Step 13: Simplify the expression:
= (cos^2A) / (1 - cos A * sin A)

As we can see, the left side of the equation (sin^3A) / (1 - cos A * sin A) is equal to the right side of the equation (cos^2A) / (1 - cos A * sin A). Therefore, the given equation is proved by using the identity sec^2A = 1 - tan^2A.