Two acute angles alpha and beta satisfy alpha + beta = 90 degrees and the ratio alpha : beta = 2 : 1. What is the simplified ratio cos alpha : cos beta?

Introduction / Context:
This trigonometry question links angle ratios with complementary angles. You must convert the given angle ratio into actual angle measures, then use standard cosine values to find the ratio cos alpha : cos beta. It tests both basic angle manipulation and knowledge of special angles.

Given Data / Assumptions:
- alpha and beta are acute angles.
- alpha + beta = 90 degrees, so they are complementary.
- The ratio alpha : beta = 2 : 1.
- We must find cos alpha : cos beta.

Concept / Approach:
From the ratio alpha : beta = 2 : 1 and the fact that alpha + beta = 90 degrees, we can treat alpha as 2k and beta as k for some positive k. Solving 2k + k = 90 gives their exact measures. Then we use well known cosine values for 60 degrees and 30 degrees and form the desired ratio.

Step-by-Step Solution:
Let alpha = 2k and beta = k for some positive k. Given alpha + beta = 90 degrees, we have 2k + k = 90, so 3k = 90. Thus k = 30 degrees. So alpha = 2k = 60 degrees and beta = k = 30 degrees. Now compute cos alpha and cos beta using special angle values. cos 60 degrees = 1 / 2. cos 30 degrees = √3 / 2. Therefore cos alpha : cos beta = (1 / 2) : (√3 / 2). Dividing both terms by 1 / 2, we get the simplified ratio 1 : √3.

Verification / Alternative check:
You can verify complementary relations by noting that cos 60 degrees equals sin 30 degrees and cos 30 degrees equals sin 60 degrees. Substituting alpha and beta into these relationships still leads to the same ratio, confirming the calculation.

Why Other Options Are Wrong:
Ratios like 1 : 3, 1 : √2, 1 : 2 or 2 : 1 come from incorrect angle values or from forgetting to simplify the ratio by cancelling the common factor 1 / 2. None of these match the exact cosine values for 60 and 30 degrees.

Common Pitfalls:
A common mistake is to misinterpret the ratio alpha : beta = 2 : 1 as alpha = 2 degrees and beta = 1 degree or to assign angles that do not sum to 90 degrees. Another pitfall is to invert the ratio and compute cos beta : cos alpha instead. Always check that your angles satisfy both the ratio and the sum condition.

Final Answer:
The required ratio is 1:√3.