Evaluate the trigonometric expression cot 60 degrees minus sec 30 degrees using exact standard values and then select the correct simplified result from the options provided.

Introduction / Context:
This problem tests your understanding of standard trigonometric values for special angles and your ability to combine them correctly in a simple expression. You are asked to evaluate cot 60 degrees minus sec 30 degrees exactly, without using decimal approximations, and then pick the correct simplified value.

Given Data / Assumptions:

• Angle 60 degrees is a standard special angle.
• Angle 30 degrees is also a standard special angle.
• cot 60 degrees = cos 60 degrees / sin 60 degrees.
• sec 30 degrees = 1 / cos 30 degrees.
• We use exact values, not calculator approximations.

Concept / Approach:
First recall the basic trigonometric values: sin 60 degrees, cos 60 degrees, and cos 30 degrees. Then compute cot 60 degrees from sine and cosine, compute sec 30 degrees as the reciprocal of cos 30 degrees, and subtract the second result from the first. Finally, simplify the expression as a single surd fraction.

Step-by-Step Solution:
Use standard values: sin 60 degrees = √3/2 and cos 60 degrees = 1/2. Compute cot 60 degrees = cos 60 degrees / sin 60 degrees = (1/2) / (√3/2) = 1/√3. Use cos 30 degrees = √3/2, so sec 30 degrees = 1 / cos 30 degrees = 1 / (√3/2) = 2/√3. Now evaluate cot 60 degrees - sec 30 degrees = 1/√3 - 2/√3 = -1/√3.
Verification / Alternative check:
You can confirm the result by rationalising denominators if desired. For example, 1/√3 = √3/3 and 2/√3 = 2√3/3, so their difference is √3/3 - 2√3/3 = -√3/3, which is equivalent to -1/√3. Both representations describe the same exact quantity.

Why Other Options Are Wrong:
Option a ( -1/3 ) ignores the square root and therefore is too small in magnitude. Option c ( 1/3 ) and option d ( 1/√3 ) have the wrong sign because they correspond to cot 60 degrees alone or a sign mistake. Option e ( -√3 ) comes from misusing tan or confusing sec and cos for 60 degrees instead of 30 degrees.

Common Pitfalls:
A common error is to mix up the values for 30 degrees and 60 degrees or to invert the wrong function. Another frequent mistake is to approximate the surds and then round, which is not required here. Always start from the exact values sin 30 degrees, cos 30 degrees, sin 60 degrees, and cos 60 degrees when working with such questions.

Final Answer:
The exact simplified value of cot 60 degrees minus sec 30 degrees is -1/√3.