In a trigonometric identity problem, if tan θ + sec θ = (x - 2) / (x + 2), determine the exact value of cos θ in terms of x (for values of x where the expressions are defined).

Introduction / Context:
This question checks your ability to connect algebraic expressions with trigonometric identities. You are given a relationship between tan θ, sec θ and an algebraic parameter x, and you must express cos θ purely in terms of x. This requires using identities that link tan θ and sec θ and then solving a small system of equations. Such problems are standard in aptitude tests and also appear in school level trigonometry exercises.

Given Data / Assumptions:

• tan θ + sec θ = (x - 2) / (x + 2), which we denote as k.
• θ is such that tan θ and sec θ are defined (cos θ is non zero).
• We must find cos θ as a rational expression in x.
• All operations use standard trigonometric identities.

Concept / Approach:
Let us denote t = tan θ and s = sec θ. Two key facts are used: 1. t + s = k, where k = (x - 2) / (x + 2). 2. sec^2 θ - tan^2 θ = 1, which gives s^2 - t^2 = 1. Now notice that (s + t)(s - t) = s^2 - t^2 = 1. Since s + t = k, it follows that s - t = 1 / k. With this, we get a simple system of two linear equations in s and t. Solving for s allows us to write sec θ in terms of x and then cos θ = 1 / s in terms of x.

Step-by-Step Solution:
Step 1: Let k = (x - 2) / (x + 2). Then s + t = k. Step 2: Using s^2 - t^2 = 1, we write (s + t)(s - t) = 1, so s - t = 1 / k. Step 3: Solve the system: s + t = k, s - t = 1 / k. Step 4: Add the equations: 2s = k + 1 / k, so s = (k + 1 / k) / 2. Step 5: Cosine is the reciprocal of secant, so cos θ = 1 / s = 2 / (k + 1 / k). Step 6: Simplify 2 / (k + 1 / k) by writing it as 2k / (k^2 + 1). Step 7: Now substitute k = (x - 2) / (x + 2). Step 8: Compute k^2 = (x - 2)^2 / (x + 2)^2 and k^2 + 1 = [(x - 2)^2 + (x + 2)^2] / (x + 2)^2 = [2x^2 + 8] / (x + 2)^2 = 2(x^2 + 4) / (x + 2)^2. Step 9: Thus cos θ = 2k / (k^2 + 1) = 2 * (x - 2)/(x + 2) * (x + 2)^2 / [2(x^2 + 4)] = (x - 2)(x + 2) / (x^2 + 4). Step 10: Multiply the numerator: (x - 2)(x + 2) = x^2 - 4. Step 11: Final expression: cos θ = (x^2 - 4) / (x^2 + 4).

Verification / Alternative check:
As a quick check, choose a convenient value of x for which all expressions are defined, compute k, then find tan θ and sec θ by solving the system numerically, and finally compute cos θ. This numerical cos θ should match the value from (x^2 - 4) / (x^2 + 4). Because the algebra is based on exact identities and not on approximations, agreement for one non trivial value strongly suggests the formula is correct for all allowed x.

Why Other Options Are Wrong:
Option a, (x^2 - 1) / (x^2 + 1), would correspond to a different relationship and does not follow from the combination of tan θ and sec θ given. Option b simplifies to (x^2 - 2) / (x^2 + 2), which also does not match the derived result. Option d explicitly uses x^2 ± 2, again inconsistent with the algebra. Option e, (x^2 + 4) / (x^2 - 4), is the reciprocal of the correct expression and would correspond to sec θ instead of cos θ. Only option c matches the algebraic derivation exactly.

Common Pitfalls:
Students often forget the identity sec^2 θ - tan^2 θ = 1 or mistakenly use tan^2 θ + 1 = sec^2 θ without rearranging it correctly. Another common error is solving the system s + t = k and s - t = 1 / k but mixing up sec θ and tan θ, leading to incorrect reciprocals. It is also easy to make algebraic mistakes when simplifying k^2 + 1. Working step by step and simplifying only at the end helps keep the process clear.

Final Answer:
Hence, the value of cos θ in terms of x is (x^2 - 4) / (x^2 + 4).