Proof:

We are given the expression: sin(theta) * sec(theta)^2 * cos(theta) * cosec(theta)^2 = sec(theta) * cosec(theta)^3

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

Simplifying the left-hand side:

Using the trigonometric identities:

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

We can rewrite the left-hand side of the equation as:

sin(theta) * (1/cos(theta))^2 * cos(theta) * (1/sin(theta))^2

Simplifying further:

sin(theta) * (1/cos(theta))^2 * cos(theta) * (1/sin(theta))^2
= sin(theta) * (1/cos(theta))^2 * cos(theta) * (1/sin(theta))^2

Since (1/cos(theta))^2 * cos(theta) = 1 (the square of the reciprocal of cos(theta) is equal to 1) and (1/sin(theta))^2 * sin(theta) = 1 (the square of the reciprocal of sin(theta) is equal to 1), we can simplify the expression to:

sin(theta) * 1 * 1

This simplifies to:

sin(theta)

Simplifying the right-hand side:

The right-hand side of the equation is sec(theta) * cosec(theta)^3

Using the trigonometric identities:

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

We can rewrite the right-hand side of the equation as:

(1/cos(theta)) * (1/sin(theta))^3

Simplifying further:

(1/cos(theta)) * (1/sin(theta))^3
= (1/cos(theta)) * 1/sin(theta))^2 * 1/sin(theta)

Since (1/cos(theta)) * (1/sin(theta))^2 = 1 (the product of the reciprocal of cos(theta) and the square of the reciprocal of sin(theta) is equal to 1), we can simplify the expression to:

1 * 1/sin(theta)

This simplifies to:

1/sin(theta)

Comparing the simplified expressions:

We have simplified the left-hand side to sin(theta) and the right-hand side to 1/sin(theta).

Since sin(theta) = 1/sin(theta) (the sine function is equal to the reciprocal of the sine function), we can conclude that the given expression is true:

sin(theta) * sec(theta)^2 * cos(theta) * cosec(theta)^2 = sec(theta) * cosec(theta)^3

Hence, the given expression is proven to be true.