Given:
- sec θ = 17/8
- θ is a positive acute angle

To find:
- The value of cosec θ

Solution:

Step 1: Recall the definitions of sec θ and cosec θ.
- sec θ = 1/cos θ
- cosec θ = 1/sin θ

Step 2: Use the given information to find the value of cos θ.
- sec θ = 17/8
- 1/cos θ = 17/8
- Cross multiply: 8 = 17cos θ
- Divide both sides by 17: cos θ = 8/17

Step 3: Use the Pythagorean identity to find the value of sin θ.
- Since θ is a positive acute angle, sin θ is positive.
- sin^2 θ + cos^2 θ = 1
- Substitute the value of cos θ: sin^2 θ + (8/17)^2 = 1
- Simplify: sin^2 θ + 64/289 = 1
- Move 64/289 to the other side: sin^2 θ = 1 - 64/289
- Simplify: sin^2 θ = (289 - 64)/289
- Calculate: sin^2 θ = 225/289
- Take the square root of both sides: sin θ = √(225/289)
- Simplify: sin θ = 15/17

Step 4: Use the values of sin θ and cos θ to find the value of cosec θ.
- cosec θ = 1/sin θ = 1/(15/17) = 17/15

Answer:
The value of cosec θ is 17/15.