For an acute angle θ where cot θ = 24/7, use a right triangle representation to calculate the exact value of sec θ and then select the correct option from the list.

Introduction / Context:
This question checks your ability to convert one trigonometric ratio into another by using a right triangle model. You are given cot θ for an acute angle θ and asked to find sec θ. This is a standard technique used in many aptitude and competitive examinations to test understanding of trigonometric relationships.

Given Data / Assumptions:

• cot θ = 24/7.
• Angle θ is acute, so all basic trigonometric ratios are positive.
• cot θ = adjacent / opposite in a right triangle.
• sec θ = hypotenuse / adjacent and is positive for acute angles.

Concept / Approach:
Interpret cot θ = 24/7 as the ratio of two sides of a right triangle. Assign adjacent side = 24k and opposite side = 7k for some positive constant k. Then apply Pythagoras theorem to find the hypotenuse. Once the hypotenuse is known in terms of k, compute sec θ as hypotenuse divided by adjacent side, which will simplify to a fraction independent of k.

Step-by-Step Solution:
Let adjacent side to angle θ be 24k and opposite side be 7k, so cot θ = 24k / 7k = 24/7. Use Pythagoras theorem: hypotenuse^2 = (24k)^2 + (7k)^2 = 576k^2 + 49k^2 = 625k^2. Therefore hypotenuse = 25k because the square root of 625 is 25. Now sec θ = hypotenuse / adjacent = 25k / 24k = 25/24.
Verification / Alternative check:
You can draw a quick sketch of the triangle with sides in the ratio 7–24–25, which is a standard Pythagorean triplet. Checking cos θ = adjacent / hypotenuse = 24/25 and then sec θ = 1 / cos θ = 25/24 confirms the result.

Why Other Options Are Wrong:
Option a ( 24/25 ) is cos θ, not sec θ. Option c ( 7/25 ) corresponds loosely to the ratio opposite / hypotenuse and is actually sin θ. Option d ( 25/7 ) arises from confusing cot θ and sec θ. Option e ( 8/25 ) has no relation to the correct Pythagorean triplet and suggests an arithmetic slip.

Common Pitfalls:
A frequent mistake is to forget that sec θ is the reciprocal of cos θ, not of cot θ. Another pitfall is misapplying Pythagoras theorem or incorrectly computing the square root of 625. Always write the triangle sides clearly and double check the triplet before finalising your answer.

Final Answer:
The exact value of sec θ when cot θ = 24/7 for an acute angle is 25/24.