Solution:

Given: tan(theta) = 1/√5

We are required to find the value of (cosec^2(theta) - sec^2(theta)) / (cosec^2(theta) * sec^2(theta))

To solve this, we will first find the values of cosec^2(theta) and sec^2(theta) using the given value of tan(theta).

Finding the values of cosec^2(theta) and sec^2(theta):

We know that cosec(theta) = 1 / sin(theta) and sec(theta) = 1 / cos(theta).

Since tan(theta) = 1/√5, we can find the value of sin(theta) and cos(theta) using the properties of trigonometric functions.

We have the identity: tan(theta) = sin(theta) / cos(theta)

Substituting the given value of tan(theta) into the identity, we get:

1/√5 = sin(theta) / cos(theta)

Cross-multiplying, we have:

cos(theta) = √5 * sin(theta)

Squaring both sides, we get:

cos^2(theta) = 5 * sin^2(theta)

Since sin^2(theta) + cos^2(theta) = 1, we can substitute the value of cos^2(theta) from the above equation:

5 * sin^2(theta) + sin^2(theta) = 1

Simplifying, we get:

6 * sin^2(theta) = 1

Dividing by 6, we get:

sin^2(theta) = 1/6

Similarly, we can find the value of cos^2(theta) using the equation:

cos^2(theta) = 5 * sin^2(theta)

Substituting the value of sin^2(theta), we get:

cos^2(theta) = 5 * (1/6)

Simplifying, we get:

cos^2(theta) = 5/6

Now, we have the values of sin^2(theta) and cos^2(theta).

Finding the value of (cosec^2(theta) - sec^2(theta)) / (cosec^2(theta) * sec^2(theta)):

To find the value of (cosec^2(theta) - sec^2(theta)) / (cosec^2(theta) * sec^2(theta)), we substitute the values of cosec^2(theta) and sec^2(theta) that we found earlier.

(cosec^2(theta) - sec^2(theta)) / (cosec^2(theta) * sec^2(theta)) = (1/sin^2(theta) - 1/cos^2(theta)) / (1/sin^2(theta) * 1/cos^2(theta))

Substituting the values, we get:

(1/(1/6) - 1/(5/6)) / (1/(1/6) * 1/(5/6))

Simplifying, we get:

(6 - 6/5) / (6/5 * 6)

= (30/5 - 6/5) / 36/5

= (24/5) / (36/5)

= 24/36

= 2/3