Question

For real x and if \(x + \frac{1}{x} = 2 \cos \theta\) then \(\cos \theta\) is

• A. \(\pm 1\)
• B. 1/2
• C. 1
• D. \(\pm 1/2\)

Hint:

This is a classic problem that tests your knowledge of the ranges of functions. Whenever you see the expression \(x + \frac{1}{x}\) for a real number \(x\), immediately recall its range: \((-\infty, -2] \cup [2, \infty)\). Comparing this with the bounded range of a trigonometric function like cosine or sine will quickly lead you to the solution at the boundary points.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We are given an equation that relates a real number x to a trigonometric function cos θ. We need to find the possible values of cos θ.

Step 2: Key Formula or Approach (Alternate Method):
Analyze the ranges of both sides separately. The LHS has a known range via AM-GM inequality, and RHS range comes from cos θ bounds. Equality only at intersection points.

Step 3: Detailed Explanation:
Equation: x + 1/x = 2 cos θ. For LHS: By AM-GM, for x>0, x+1/x ≥ 2. For x<0, x+1/x ≤ -2. So |LHS| ≥ 2. For RHS: cos θ ∈ [-1, 1], so 2 cos θ ∈ [-2, 2]. So |RHS| ≤ 2. Equality possible only when LHS = RHS = 2 or LHS = RHS = -2. 2 cos θ = 2 → cos θ = 1. 2 cos θ = -2 → cos θ = -1.

Step 4: Final Answer:
The only possible values for cos θ are 1 and -1.