If sec A + tan A = a for an acute angle A, what is the value of cos A in terms of a?

Introduction / Context:
This trigonometric problem involves a standard technique of using the relation between sec A and tan A when their sum is known. You are given sec A + tan A = a and need to find cos A. This type of question checks your understanding of how to manipulate reciprocal and Pythagorean identities without needing the explicit value of the angle A.

Given Data / Assumptions:

• sec A + tan A = a.
• A is an acute angle, so cos A and sec A are positive.
• We need cos A expressed in terms of a.

Concept / Approach:
A key idea is that sec A − tan A is the reciprocal of sec A + tan A for any angle where these functions are defined. Specifically, (sec A + tan A)(sec A − tan A) = 1. Once sec A and sec A − tan A are expressed in terms of a, we can solve for sec A and then invert to get cos A.

Step-by-Step Solution:
Given sec A + tan A = a. Use the identity (sec A + tan A)(sec A − tan A) = sec^2 A − tan^2 A. Recall that sec^2 A − tan^2 A = 1. Therefore (sec A + tan A)(sec A − tan A) = 1. Given sec A + tan A = a, we get a(sec A − tan A) = 1. So sec A − tan A = 1/a. Now add the two equations: (sec A + tan A) + (sec A − tan A) = a + 1/a. Left side simplifies to 2 sec A. So 2 sec A = a + 1/a. Therefore sec A = (a + 1/a) / 2. Now cos A = 1 / sec A = 1 / [(a + 1/a)/2] = 2 / (a + 1/a). Simplify the denominator: a + 1/a = (a^2 + 1)/a. Thus cos A = 2 / ((a^2 + 1)/a) = 2a / (a^2 + 1).
Verification / Alternative check:
You can choose a specific acute angle, for example A = 45°. Then sec 45° + tan 45° = √2 + 1, so a = √2 + 1. Compute 2a/(a^2 + 1) and verify that it equals cos 45° = √2/2, confirming the formula.

Why Other Options Are Wrong:
The expression (a^2 + 1)/(2a) is actually sec A, not cos A. The form (a^2 − 1)/(2a) relates to tan A when expressed in terms of a. Options involving a^2 − 1 in the denominator do not fit the identity used and lead to incorrect values when tested with specific angles.

Common Pitfalls:
Many students forget the identity sec^2 A − tan^2 A = 1 and instead attempt more complicated algebra. Another common mistake is inverting a + 1/a incorrectly when computing cos A. Carefully applying the identity and doing the fraction simplification step by step prevents these errors.

Final Answer:
The value of cos A is 2a / (a^2 + 1).