What is the exact simplified value of the trigonometric expression cot 45° + (1/3) cosec 60°, expressed in standard surd form?

Introduction / Context:
This question checks basic trigonometric values for standard angles and the ability to combine them into a single simplified expression. Values at 45 degrees and 60 degrees are frequently tested because they produce neat surds involving sqrt(2) and sqrt(3). The goal is to simplify cot 45° and cosec 60° and then combine them carefully into one fraction in exact form.

Given Data / Assumptions:

• The expression is cot 45° + (1/3) cosec 60°.
• We work in degrees.
• We use standard exact values for trigonometric functions at 45° and 60°.
• We are required to express the answer in exact surd form without decimal approximation.

Concept / Approach:
Recall the standard values: tan 45° = 1, so cot 45° = 1 / tan 45° = 1. For 60 degrees, sin 60° = sqrt(3) / 2, so cosec 60° = 1 / sin 60° = 2 / sqrt(3). Substitute these values into the expression and then simplify systematically. To present the result cleanly, rationalise the denominator where necessary and combine the constant 1 with the surd term over a common denominator.

Step-by-Step Solution:
Compute cot 45°. Since tan 45° = 1, cot 45° = 1.Compute cosec 60°. Since sin 60° = sqrt(3) / 2, cosec 60° = 1 / sin 60° = 2 / sqrt(3).Now evaluate (1/3) cosec 60°: (1/3) * (2 / sqrt(3)) = 2 / (3 sqrt(3)).So the expression becomes 1 + 2 / (3 sqrt(3)).Rationalise the small fraction: 2 / (3 sqrt(3)) = (2 sqrt(3)) / (9).Thus the total is 1 + (2 sqrt(3) / 9) = (9 / 9) + (2 sqrt(3) / 9) = (9 + 2 sqrt(3)) / 9.

Verification / Alternative check:
We can check numerically using approximate values. Take sqrt(3) approximately equal to 1.732. Then sin 60° is about 0.866, cosec 60° is about 1.155, and (1/3) cosec 60° is about 0.385. Adding 1 gives approximately 1.385. Now evaluate (9 + 2 sqrt(3)) / 9 numerically: 2 sqrt(3) is about 3.464, so 9 + 3.464 ≈ 12.464. Divide by 9 to get about 1.3849, which matches our previous estimate, confirming the correctness.

Why Other Options Are Wrong:

• √3 and √3 + 2 are much larger and do not match the approximate value near 1.385.
• (2√2 + 3)/√6 does not simplify to the same numerical value when computed.
• (3 + √3)/3 gives roughly 1.244, which is different from 1.385.
• Only (9 + 2√3)/9 agrees with both exact simplification and numerical checking.

Common Pitfalls:

• Using wrong standard values for sine or cosine at 60 degrees or for tangent at 45 degrees.
• Forgetting to multiply cosec 60° by 1/3 carefully, which alters the final answer.
• Not rationalising the denominator correctly and therefore failing to match any of the given options.

Final Answer:
(9 + 2√3)/9