Proof of the Identity:

To prove the identity, we need to manipulate the left-hand side of the equation until it becomes equal to the right-hand side.

Step 1: Simplify the left-hand side using the identity (a-b)(a+b) = a^2 - b^2.

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

Step 2: Use the identity cos^2A + sin^2A = 1 to simplify the numerator.

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

Step 3: Use the identity 1/cosA = secA and sinA/cosA = tanA to further simplify the expression.

(1-sin^2A) / (cosA-sinA) = [(1/sin^2A) - 1] / [(1/cosA) - (sinA/cosA)]
= [(1/sin^2A) - 1] / [secA - tanA]

Step 4: Use the identity 1/sinA = cosecA and 1/tanA = cotA to get the final expression.

[(1/sin^2A) - 1] / [secA - tanA] = [(cosec^2A) - 1] / [secA - (1/cotA)]
= cosecA cotA

Therefore, the left-hand side of the identity is equal to the right-hand side, and the identity is proven.

LHS=cosA-sinA+1/cosA+sinA-1=(cosA/cosA-sinA/sinA+1/sinA)/cosA/sinA+sinA/sinA-1/sinA(by dividing d^r and n^r both by cosA)=(cotA+cosecA-1)/(cotA-cosecA+1)(cotA+cosecA)-(cosec^2A-cot^2A)/cotA-cosecA+1=(cotA+cosecA)-(cosecA+cotA)(cosecA-cotA)/cotA-cosecA+1=(cotA+cosecA)(1-cosecA+cotA)/(cotA-cosecA+1)=cotA+cosecAHENCE. PROVEDfirst of all kahi likh kr dekh lo bhaii phirr samajh me aayegaso,its good for you to write it firstly........