What is the exact value of the expression (1/3 - cot 60°) expressed in simplest surd form?

Introduction / Context:
This is a straightforward trigonometry and simplification problem. It checks your knowledge of standard trigonometric values for special angles and your ability to simplify a numerical expression involving a trigonometric ratio and a simple fraction, presenting the result in exact surd form rather than as a decimal.

Given Data / Assumptions:

• Expression: 1/3 - cot 60°.
• Angles are in degrees.
• Required: an exact simplified value using surds.

Concept / Approach:
We recall the standard value of tan 60°, and use it to find cot 60°. Then we subtract this value from 1/3 and simplify the resulting algebraic expression. Since the result involves sqrt(3), we express the answer in a rationalised and simplified form that can be matched with one of the given surd options.

Step-by-Step Solution:
Step 1: Recall that tan 60° = sqrt(3). Step 2: Since cot θ is the reciprocal of tan θ, cot 60° = 1 / tan 60° = 1 / sqrt(3). Step 3: Substitute this into the expression: 1/3 - cot 60° = 1/3 - 1/sqrt(3). Step 4: To combine the terms, use a common denominator of 3sqrt(3). Step 5: Rewrite 1/3 as sqrt(3)/(3sqrt(3)). Step 6: Rewrite 1/sqrt(3) as 3/(3sqrt(3)). Step 7: Subtract to get [sqrt(3) - 3]/(3sqrt(3)). Step 8: Factor a negative sign if desired: [sqrt(3) - 3]/(3sqrt(3)) = (1 - sqrt(3))/3 after simplifying the surd forms.

Verification / Alternative check:
Numerically, tan 60° is approximately 1.732, so cot 60° is about 0.577. Then 1/3 is approximately 0.333. So 1/3 - cot 60° is roughly 0.333 - 0.577 = -0.244. Evaluating (1 - sqrt(3))/3 numerically gives (1 - 1.732)/3 ≈ -0.244, confirming that the exact surd form matches the decimal approximation.

Why Other Options Are Wrong:
(2 - sqrt(3))/(2sqrt(3)) and (sqrt(3) - 4)/(2sqrt(3)) represent different combinations of sqrt(3) that do not simplify to the same numeric value as 1/3 - cot 60°. (sqrt(2) - 1)/sqrt(2) is based on the angle 45° rather than 60°. (sqrt(3) - 1)/3 has the opposite sign from the correct answer and approximates to a positive value, which does not match the negative result of the original expression.

Common Pitfalls:
A frequent mistake is to treat cot 60° as sqrt(3) instead of 1/sqrt(3), confusing it with tan 60°. Another common error is failing to use a common denominator when combining the fractions, which leads to incorrect simplification. Careful handling of surds and denominators avoids these issues.

Final Answer:
The exact value of 1/3 - cot 60° is (1 - sqrt(3))/3.