Evaluate the trigonometric expression Cot 60° minus Cos 45°. Which of the following simplified forms is correct?

Introduction / Context:
This problem tests basic values of trigonometric functions at standard angles and your ability to simplify an expression involving those functions. You are asked to evaluate Cot 60° minus Cos 45° and match the result to one of the given algebraic expressions involving square roots.

Given Data / Assumptions:

• Angle values: 60° and 45°.
• We work with standard trigonometric values.
• Expression to evaluate: Cot 60° - Cos 45°.

Concept / Approach:
Use the known exact values for trigonometric functions at special angles. For 60°, Cos 60° = 1/2, Sin 60° = √3/2, so Cot 60° = Cos 60° / Sin 60° = (1/2) / (√3/2) = 1 / √3. For 45°, Cos 45° = √2 / 2. Substitute these values and simplify the resulting expression to match one of the options.

Step-by-Step Solution:
Step 1: Write Cot 60° in terms of sine and cosine: Cot 60° = Cos 60° / Sin 60°.Step 2: Substitute standard values: Cos 60° = 1/2 and Sin 60° = √3 / 2, so Cot 60° = (1/2) / (√3 / 2) = 1 / √3.Step 3: Write Cos 45° = √2 / 2.Step 4: The expression becomes 1 / √3 - √2 / 2.Step 5: Simplify this difference to the forms given in the options. One way is to rationalise and combine over a common denominator, or compare numerically with each option.Step 6: Evaluating numerically, 1 / √3 - √2 / 2 is approximately -0.1298. Evaluating each option shows that only (√2 - √3) / √6 gives the same approximate numerical value, because (√2 - √3) / √6 is also approximately -0.1298.

Verification / Alternative check:
For a purely algebraic verification, you can write 1 / √3 as √3 / 3 and √2 / 2 as √6 / (2√3). Then express everything with denominator 2√3 and compare to the simplified form (√2 - √3) / √6, which after rationalisation matches the same expression. This confirms that the two forms are equivalent.

Why Other Options Are Wrong:
The expressions (9 - 2√3)/9, (2√6 - 1)/√3 and (1 - 2√3)/2 represent different numerical values that do not equal 1 / √3 - √2 / 2. Option (√3 - √2)/√6 has the opposite sign to the correct result and therefore does not match the evaluated expression.

Common Pitfalls:
Common mistakes include using incorrect standard values, such as confusing Sin 60° and Sin 30°, or forgetting that Cot 60° is the reciprocal of Tan 60°. Another error is careless simplification or failing to check that the sign of the final result matches the expression. Careful substitution and comparison prevent these issues.

Final Answer:
The correct simplified value of Cot 60° − Cos 45° is (√2 - √3)/√6.