In trigonometry, let θ be an acute angle with Sec θ = 13/12. Using a right triangle representation, determine the exact value of Cot θ in simplest fractional form.

Introduction / Context:
This question involves converting one trigonometric ratio into another by using right triangle definitions. You are given the value of sec θ for an acute angle and asked to find cot θ. Such conversions are routine in trigonometry and are often tested to ensure you can move between different trigonometric functions using geometric reasoning or identities.

Given Data / Assumptions:

• θ is an acute angle, so 0° < θ < 90° and all primary trigonometric ratios are positive.
• Sec θ = 13/12.
• We need to find Cot θ, expressed as a simplified fraction.
• Standard right triangle definitions of sine, cosine, tangent, and cotangent apply.

Concept / Approach:
By definition, sec θ is the reciprocal of cos θ, so sec θ = 1 / cos θ. Given sec θ, we can find cos θ by inverting the fraction. Once cos θ is known, we can imagine a right triangle in which the adjacent side and hypotenuse reflect this ratio. Using Pythagoras theorem, we find the remaining side, which allows us to compute sin θ and then cot θ = cos θ / sin θ or cot θ = adjacent / opposite directly. Because θ is acute, all ratios are positive, which simplifies the interpretation.

Step-by-Step Solution:
1) Start from the relation sec θ = 13/12. 2) Use the reciprocal relation sec θ = 1 / cos θ to find cos θ = 12 / 13. 3) Interpret cos θ = adjacent / hypotenuse in a right triangle, so take the adjacent side as 12 units and the hypotenuse as 13 units. 4) Use Pythagoras theorem to find the opposite side: opposite^2 = hypotenuse^2 − adjacent^2 = 13^2 − 12^2 = 169 − 144 = 25. 5) Therefore the opposite side has length √25 = 5. 6) Now cot θ is defined as adjacent / opposite, or cot θ = cos θ / sin θ. 7) Using the triangle sides, cot θ = 12 / 5.

Verification / Alternative check:
We can double check by computing sin θ and verifying the Pythagoras identity. From the triangle, sin θ = opposite / hypotenuse = 5 / 13 and cos θ = 12 / 13. Check that sin^2 θ + cos^2 θ = (5/13)^2 + (12/13)^2 = 25/169 + 144/169 = 169/169 = 1, confirming the triangle is consistent. Then cot θ = cos θ / sin θ = (12/13) / (5/13) = 12/5. This matches the value obtained from adjacent / opposite directly.

Why Other Options Are Wrong:
Option b (13/5) and option e (12/13) mis place the roles of the sides or ratios. Option c (5/12) is the reciprocal of tan θ, if tan θ were 12/5, but here tan θ is actually 5/12. Option d (5/13) corresponds to sin θ, not cot θ. Only option a, 12/5, correctly represents the cotangent ratio for the given sec θ value.

Common Pitfalls:
Mistakes often occur when students invert the ratio incorrectly, for example writing cos θ = 13/12 instead of 12/13. Another frequent error is confusing tan θ with cot θ or forgetting that cot θ = adjacent / opposite, not hypotenuse over adjacent. Carefully setting up the right triangle and labelling the sides according to cos θ = adjacent / hypotenuse prevents these misunderstandings and makes the computation straightforward.

Final Answer:
For an acute angle θ with Sec θ = 13/12, the value of Cot θ is 12/5.