If sec θ (cos θ + sin θ) = √2, then what is the value of (2 sin θ) / (cos θ – sin θ)?

Introduction / Context:
This question checks skill in manipulating trigonometric expressions with secant, sine, and cosine. A relationship is given involving sec θ and a sum of sine and cosine, and we are asked to evaluate another expression involving sine and cosine. The trick is to express everything in terms of tangent and then simplify.

Given Data / Assumptions:

• sec θ (cos θ + sin θ) = √2.
• We need to find the value of (2 sin θ) / (cos θ - sin θ).
• θ is an angle for which cos θ is not zero, because sec θ is defined.

Concept / Approach:
We express sec θ as 1 / cos θ, which instantly simplifies sec θ (cos θ + sin θ). This leads to a simple equation involving tan θ. Once we have tan θ, the target expression can be written in terms of tan θ using sin θ = tan θ cos θ. Algebra then gives a numerical value, which we can simplify to a simple surd.

Step-by-Step Solution:
Start from sec θ (cos θ + sin θ) = √2. Write sec θ as 1 / cos θ. Then we have (1 / cos θ) (cos θ + sin θ) = √2. Simplify the left side: (cos θ + sin θ) / cos θ = 1 + tan θ. So 1 + tan θ = √2, which gives tan θ = √2 - 1. Now consider the required expression E = (2 sin θ) / (cos θ - sin θ). Express sin θ as tan θ cos θ. Then E = [2 tan θ cos θ] / [cos θ - tan θ cos θ]. Factor cos θ in the denominator: cos θ - tan θ cos θ = cos θ (1 - tan θ). Thus E = [2 tan θ cos θ] / [cos θ (1 - tan θ)] = 2 tan θ / (1 - tan θ). Substitute tan θ = √2 - 1. So E = 2(√2 - 1) / [1 - (√2 - 1)]. Simplify the denominator: 1 - (√2 - 1) = 2 - √2. So E = 2(√2 - 1) / (2 - √2). Multiply numerator and denominator by the conjugate (2 + √2). E = [2(√2 - 1)(2 + √2)] / [(2 - √2)(2 + √2)] = [2( (√2 - 1)(2 + √2) )] / (4 - 2). Compute 4 - 2 = 2. So E = (2 / 2) * (√2 - 1)(2 + √2) = (√2 - 1)(2 + √2). Expand: (√2 - 1)(2 + √2) = 2√2 + 2 - √2 - 1 = √2 + 1. But from 1 + tan θ = √2 and tan θ = √2 - 1, we know √2 = tan θ + 1, which equals √2. Evaluating the numeric value more directly gives E = √2.

Verification / Alternative check:
We can pick an angle θ that satisfies tan θ = √2 - 1 numerically and compute both sides using a calculator. Substituting this θ into (2 sin θ) / (cos θ - sin θ) yields a value equal to √2 within rounding error, confirming the algebraic result.

Why Other Options Are Wrong:
1/√2 and 3/√2: These come from incomplete simplification of the surd expression or from mistakes in rationalizing the denominator.
3√2 and 2√2: These are larger values that would result from arithmetic errors, such as adding terms incorrectly during expansion.

Common Pitfalls:
Common mistakes include forgetting that sec θ is 1 / cos θ, mishandling the algebra around tan θ, or making errors when rationalizing the denominator. Working step by step and keeping track of tan θ helps keep everything consistent and accurate.

Final Answer:
The value of (2 sin θ) / (cos θ - sin θ) is √2.