In trigonometry, find the exact value of the expression cot 60° minus sec 45°.

Introduction / Context:
This question tests basic trigonometric values of standard angles and the ability to combine them into a single exact expression. Rather than approximating the trigonometric ratios numerically, the goal is to manipulate them in radical form and match the result with one of the given options. Such questions are common in aptitude and competitive exams where quick recall of standard angle values like 30°, 45° and 60° is essential.

Given Data / Assumptions:

• We need to evaluate cot 60° − sec 45° exactly.
• Standard angle values are used: 60° and 45°.
• Angles are in degrees and lie in the first quadrant, so all trigonometric ratios are positive.

Concept / Approach:
The main ideas are:

• Use standard trigonometric values: tan 60°, cos 45° and their reciprocals.
• Recall that cot θ = 1 / tan θ and sec θ = 1 / cos θ.
• Express both terms in radical form and then combine them into a single simplified expression.
• Optionally, rationalise or rearrange to match one of the answer choices.

Step-by-Step Solution:
Step 1: Recall that tan 60° = √3, so cot 60° = 1 / tan 60° = 1 / √3 = √3 / 3. Step 2: Recall that cos 45° = √2 / 2, so sec 45° = 1 / cos 45° = 1 / (√2 / 2) = 2 / √2 = √2. Step 3: Substitute these into the expression: cot 60° − sec 45° = (√3 / 3) − √2. Step 4: To compare with options, write this over a common denominator of 3: (√3 / 3) − √2 = (√3 / 3) − (3√2 / 3). Step 5: Combine the numerators: (√3 − 3√2) / 3. Step 6: Therefore the exact value is (√3 − 3√2) / 3.
Verification / Alternative check:
Step 1: Use approximate values √2 ≈ 1.414 and √3 ≈ 1.732. Step 2: Compute cot 60° ≈ 0.577 and sec 45° ≈ 1.414, so cot 60° − sec 45° ≈ 0.577 − 1.414 ≈ −0.837. Step 3: Evaluate (√3 − 3√2) / 3 numerically: numerator ≈ 1.732 − 4.242 = −2.510; divide by 3 gives ≈ −0.837, which agrees with the direct calculation.
Why Other Options Are Wrong:
Option (√2 - √3)/√6 is roughly −0.13, which does not match the approximate value −0.84. Option (1 - 2√2)/2 is approximately −0.91, again not equal to −0.84. Option (1 - √3)/2 is about −0.37, which is far from the correct value. Option (3√2 - √3)/3 is the negative of the correct numerator sign pattern and is positive, not negative.
Common Pitfalls:
One common mistake is misremembering cot 60° as √3 instead of 1 / √3. Another error is to use sec 45° = 1 / (1 / √2) incorrectly and leave it as 2 / √2 without simplifying to √2. Students sometimes try to compare options numerically without first simplifying, which can take longer and lead to rounding errors.
Final Answer:
The exact value of cot 60° − sec 45° is (√3 - 3√2)/3.