**Proof:**

To prove: cosec(A)/sec(A) - sec(A)/cosec(A) = 2cosec^3(A)(sec(A) - 1)

We will start by simplifying the left-hand side of the equation.

Let's express cosec(A) and sec(A) in terms of sin(A) and cos(A):

cosec(A) = 1/sin(A)

sec(A) = 1/cos(A)

Now let's substitute these expressions into the left-hand side of the equation:

cosec(A)/sec(A) - sec(A)/cosec(A) = (1/sin(A))/(1/cos(A)) - (1/cos(A))/(1/sin(A))

Using the reciprocal property of division, we can rewrite this as:

= (1/sin(A)) * (cos(A)/1) - (1/cos(A)) * (sin(A)/1)

Simplifying further, we get:

= cos(A)/sin(A) - sin(A)/cos(A)

Now let's find a common denominator and combine the fractions:

= (cos^2(A) - sin^2(A))/(sin(A) * cos(A))

Using the trigonometric identity cos^2(A) - sin^2(A) = cos(2A), we can rewrite the numerator:

= cos(2A)/(sin(A) * cos(A))

Now let's simplify the right-hand side of the equation.

= 2cosec^3(A)(sec(A) - 1)

= 2(1/sin(A))^3 (1/cos(A) - 1)

= 2(1/sin(A))^3 (cos(A)/cos(A) - 1)

= 2(1/sin(A))^3 ((cos(A) - 1 * cos(A))/cos(A))

= 2(1/sin(A))^3 (cos(A) - cos^2(A))/cos(A)

Using the trigonometric identity cos^2(A) = 1 - sin^2(A), we can rewrite the numerator:

= 2(1/sin(A))^3 (cos(A) - (1 - sin^2(A)))/cos(A)

= 2(1/sin(A))^3 (cos(A) - 1 + sin^2(A))/cos(A)

= 2(1/sin(A))^3 (cos(A) - 1 + 1 - cos^2(A))/cos(A)

= 2(1/sin(A))^3 (2cos(A) - cos^2(A))/cos(A)

= 4(1/sin(A))^3 (cos(A) - cos^2(A))/cos(A)

= 4(1/sin(A))^3 (cos(A)(1 - cos(A)))/cos(A)

= 4(1/sin(A))^3 (1 - cos(A))

Now we can see that the right-hand side of the equation is equivalent to the left-hand side, thus proving the given equation.