**Proof:**

To prove the given equation l^2m^2(l^2 + m^2 + 3) = 1, we will start by expressing the left-hand side of the equation in terms of trigonometric functions using the given values cosecA - sinA = l and secA - cosA = m.

Let's begin the proof by expressing the left-hand side of the equation in terms of trigonometric functions.

**1. Expressing l and m in terms of trigonometric functions:**

From the given equations cosecA - sinA = l and secA - cosA = m, we can rewrite them as:

cosecA = l + sinA ...(1)

secA = m + cosA ...(2)

Now, let's express l and m in terms of trigonometric functions using these equations:

l = cosecA - sinA ...(3)

m = secA - cosA ...(4)

**2. Expressing the left-hand side of the equation using l and m:**

Now, let's substitute the expressions for l and m in terms of trigonometric functions into the given equation l^2m^2(l^2 + m^2 + 3) = 1:

(l^2)(m^2)(l^2 + m^2 + 3) = 1 ...(5)

Substituting the expressions for l and m from equations (3) and (4) into equation (5), we get:

[(cosecA - sinA)^2][(secA - cosA)^2][(cosecA - sinA)^2 + (secA - cosA)^2 + 3] = 1

Now, let's simplify this equation further.

**3. Simplifying the equation:**

Expanding the squares and simplifying, we get:

[(cosec^2A - 2sinA cosecA + sin^2A)][(sec^2A - 2cosA secA + cos^2A)][(cosec^2A - 2sinA cosecA + sin^2A) + (sec^2A - 2cosA secA + cos^2A) + 3] = 1

Using the trigonometric identities cosec^2A = 1 + cot^2A and sec^2A = 1 + tan^2A, we can rewrite the equation as:

[(1 + cot^2A - 2sinA cosecA + sin^2A)][(1 + tan^2A - 2cosA secA + cos^2A)][(1 + cot^2A - 2sinA cosecA + sin^2A) + (1 + tan^2A - 2cosA secA + cos^2A) + 3] = 1

Now, let's simplify further.

**4. Simplifying further:**

Expanding and simplifying the equation, we get:

[(cot^2A - 2sinA cosecA + sin^2A + 1)][(1 + tan^2A - 2cosA secA + cos^2A)][(cot^2A - 2sinA cosecA + sin^2A + 1) + (1