If cot θ = 21/20 for an acute angle θ in a right-angled triangle, what is the exact value of sec θ expressed as a rational fraction?

Introduction / Context:
This problem tests the relationship among the six basic trigonometric ratios and how to move from one ratio to another using right triangle geometry and the Pythagorean theorem. Given cot θ, you are asked to find sec θ. This involves interpreting cot θ as a ratio of sides in a right triangle and then computing cos θ and its reciprocal, sec θ.

Given Data / Assumptions:

• cot θ = 21/20.
• θ is an acute angle, so all basic trigonometric ratios are positive.
• We consider a right triangle where the angle θ is one of the acute angles.
• We must determine sec θ as a simplified fraction.

Concept / Approach:
By definition, cot θ = adjacent / opposite in a right triangle. So if cot θ = 21/20, we can take the side adjacent to θ as 21 units and the side opposite θ as 20 units. Then we use the Pythagorean theorem to find the hypotenuse. Once we know all three sides, cos θ = adjacent / hypotenuse, and sec θ is its reciprocal, hypotenuse / adjacent. The numbers 20, 21, and 29 form a well known Pythagorean triple because 20^2 + 21^2 = 29^2.

Step-by-Step Solution:
Interpret cot θ = 21/20 as adjacent side = 21 and opposite side = 20.Use Pythagoras theorem to find the hypotenuse h: h^2 = 21^2 + 20^2.Compute 21^2 = 441 and 20^2 = 400, so h^2 = 441 + 400 = 841.Thus h = sqrt(841) = 29.Now cos θ = adjacent / hypotenuse = 21 / 29.Secant is the reciprocal of cosine: sec θ = 1 / cos θ = 29 / 21.

Verification / Alternative check:
We can verify by computing cot θ and sec θ directly from the sides. Cot θ = adjacent / opposite = 21 / 20, which matches the given ratio. Sec θ = hypotenuse / adjacent = 29 / 21. Also check the identity 1 + tan^2 θ = sec^2 θ. Here tan θ = opposite / adjacent = 20 / 21, so tan^2 θ = 400 / 441. Then sec^2 θ should be 1 + 400 / 441 = 841 / 441, and sec θ is sqrt(841 / 441) = 29 / 21, consistent with our result.

Why Other Options Are Wrong:

• 21/29 is cos θ, not sec θ.
• 29/20 and 20/29 correspond to using the wrong pair of sides to form the ratio.
• 20/21 equals tan θ, not sec θ.
• Only 29/21 correctly represents the reciprocal of cos θ.

Common Pitfalls:

• Confusing cot θ with tan θ and swapping the opposite and adjacent sides.
• Forgetting that sec θ is the reciprocal of cos θ and instead dividing in the wrong order.
• Making arithmetic mistakes while computing the hypotenuse, especially when squaring and adding 20 and 21.

Final Answer:
29/21