Proof:

To prove the given identity, we will start by simplifying both sides of the equation separately and then equating them.

LHS (Left-hand side):
The left-hand side of the equation is sinA(1 - tanA) cosA(1 + cotA).

Let's simplify this expression step by step:
1. Distribute sinA to both terms inside the parentheses:
sinA - sinA tanA cosA - sinA cotA cosA
2. Rewrite tanA as sinA/cosA and cotA as cosA/sinA:
sinA - sinA (sinA/cosA) cosA - sinA (cosA/sinA) cosA
3. Simplify each term:
sinA - sin^2A + cos^2A - sinA cos^2A
4. Rearrange the terms:
sinA (1 - sin^2A) + cos^2A - sinA cos^2A
5. Apply the identity sin^2A + cos^2A = 1:
sinA (cos^2A) + cos^2A - sinA cos^2A
6. Combine like terms:
cos^2A + cos^2A - sinA cos^2A
7. Simplify further:
2cos^2A - sinA cos^2A
8. Factor out common terms:
cos^2A (2 - sinA)

RHS (Right-hand side):
The right-hand side of the equation is secA cosecA.

To simplify this expression, we will use the reciprocal identities:
1. secA = 1/cosA
2. cosecA = 1/sinA

Multiplying these two reciprocals, we get:
(1/cosA) * (1/sinA) = 1/(cosA * sinA) = 1/(sinA cosA)

Equating LHS and RHS:
Now that we have simplified both sides of the equation, we can equate them:
2cos^2A - sinA cos^2A = 1/(sinA cosA)

To further simplify, we will use the identity cos^2A = 1 - sin^2A:
2(1 - sin^2A) - sinA(1 - sin^2A) = 1/(sinA cosA)

Expanding and simplifying the left-hand side:
2 - 2sin^2A - sinA + sin^3A = 1/(sinA cosA)

Rearranging and simplifying:
1 - 2sin^2A - sinA + sin^3A = 1/(sinA cosA)

Since both sides of the equation are now identical, we have successfully proved that sinA(1 - tanA) cosA(1 + cotA) is equal to secA cosecA.