Using standard trigonometric values in degrees, evaluate the expression (1 / √2) · Cot 45° + (1 / √3) · Cosec 60°. What is the exact simplified value of this trigonometric sum?

Introduction / Context:
This question tests your ability to work with standard trigonometric values and surds, and to simplify expressions involving square roots. Such problems are common in quantitative aptitude tests, where exact simplification is more important than decimal approximations.

Given Data / Assumptions:

• Angles are measured in degrees.
• We must compute (1 / √2) * Cot 45° + (1 / √3) * Cosec 60°.
• Standard values at 45° and 60° are assumed known.

Concept / Approach:
We recall that Cot 45° = 1 and Sin 60° = √3 / 2. Since Cosec 60° is the reciprocal of Sin 60°, we can write Cosec 60° = 2 / √3. Then we substitute these values into the expression, simplify each term carefully, and combine them into a single simplified surd expression.

Step-by-Step Solution:
Start with (1 / √2) * Cot 45° + (1 / √3) * Cosec 60°. Use Cot 45° = 1, so the first term becomes 1 / √2. Use Sin 60° = √3 / 2, hence Cosec 60° = 1 / (√3 / 2) = 2 / √3. The second term is (1 / √3) * (2 / √3) = 2 / 3. So the expression becomes 1 / √2 + 2 / 3. Take a common denominator 3√2: 1 / √2 = (3) / (3√2) and 2 / 3 = (2√2) / (3√2). Add them: (3 + 2√2) / (3√2).
Verification / Alternative check:
Approximate numerically: 1 / √2 is about 0.707 and 2 / 3 is about 0.667. Their sum is roughly 1.374. The expression (3 + 2√2) / (3√2) gives a similar decimal value when evaluated, confirming the algebraic simplification is correct.

Why Other Options Are Wrong:
Option a and option c are alternative surd forms that do not simplify to the same numeric value. Option b (√3) is about 1.732, which is significantly larger than the correct decimal value near 1.374. Option e also does not equal the computed value when simplified.

Common Pitfalls:
A frequent error is to mix up Cosec 60° with Cosec 30°, or to confuse Cot and Tan at 45°. Another pitfall is mishandling the surd rationalisation step, which can lead to incorrect numerators or denominators. Working systematically with common denominators helps keep the simplification correct.

Final Answer:
The exact value of the expression is (3 + 2√2) / (3√2).