In right triangle ΔXYZ, which is right angled at Y, suppose that ∠Z = 60°. Let X be the remaining acute angle. Evaluate the expression (1 / √2) × Sec X and choose its exact value from the options given.

Introduction / Context:
This question combines basic geometry of right triangles with trigonometric ratios. You must use the fact that the sum of angles in a triangle is 180 degrees and then apply standard trigonometric values for well known angles.

Given Data / Assumptions:

• ΔXYZ is a right triangle right angled at Y.
• ∠Z = 60°.
• Therefore the remaining acute angle at X satisfies ∠X + 60° + 90° = 180°.
• We need to compute (1 / √2) * Sec X.

Concept / Approach:
In any triangle, the sum of the three interior angles is 180°. For a right triangle, one angle is 90°, so the two acute angles must add to 90°. Once we find ∠X, we use standard trigonometric values: Sec θ is the reciprocal of Cos θ. Finally we multiply Sec X by 1 / √2 and simplify the resulting surd expression.

Step-by-Step Solution:
Start with the angle sum: ∠X + ∠Y + ∠Z = 180°. Given ∠Y = 90° and ∠Z = 60°, so ∠X + 90° + 60° = 180°. Thus ∠X + 150° = 180°, so ∠X = 30°. Now use the standard value Cos 30° = √3 / 2. Sec 30° is 1 / Cos 30° = 1 / (√3 / 2) = 2 / √3. We must compute (1 / √2) * Sec X = (1 / √2) * (2 / √3). Multiply the numerators and denominators: (1 * 2) / (√2 * √3) = 2 / √6.
Verification / Alternative check:
You can approximate the values numerically. Cos 30° is about 0.866, so Sec 30° is approximately 1.155. Then (1 / √2) is about 0.707, and the product is roughly 0.707 * 1.155 which is about 0.817. The fraction 2 / √6 is also approximately 0.816, confirming the result.

Why Other Options Are Wrong:
Option a (2 / √3) is Sec 30° alone without the factor 1 / √2. Option b (1 / √6) and option c (1 / √3) are too small compared with the approximate numeric result. Option e (√2) is larger than 1 and does not match the computed value.

Common Pitfalls:
A frequent mistake is to assume X is 60° rather than 30°, ignoring the right angle at Y. Another pitfall is confusing Sec with Cos or using the wrong special angle values. Always check the triangle angle sum carefully before applying trigonometric ratios.

Final Answer:
The value of (1 / √2) × Sec X is 2 / √6.