In right-angled triangle ABC, right angled at B, if ∠C = 60° (so ∠A = 30°), what is the simplified exact value of the expression (cot A − 1/2)?

Introduction / Context:
This trigonometry question uses a standard right triangle with angles 30 degrees, 60 degrees, and 90 degrees. You are asked to evaluate an expression involving cot A when angle C is 60 degrees. Once the relationship between angles is clear, the problem reduces to using well known trigonometric ratios for 30 degrees and then subtracting a fraction.

Given Data / Assumptions:

• Triangle ABC is right angled at B.
• Angle C = 60°, so angle A = 30°.
• The expression to evaluate is cot A − 1/2.
• We use standard exact values of trigonometric ratios for 30 degrees.

Concept / Approach:
For angle A = 30 degrees in a 30°–60°–90° triangle, the sides can be taken in the ratio 1 : √3 : 2, where 1 is opposite 30 degrees, √3 is opposite 60 degrees, and 2 is the hypotenuse. Cotangent is defined as adjacent / opposite. For a 30 degree angle, cot 30° equals √3. Once cot A = √3 is known, we simply compute √3 − 1/2 as the required expression.

Step-by-Step Solution:
Since triangle ABC is right angled at B and angle C = 60°, the remaining acute angle A must be 30°.In a 30°–60°–90° triangle, let the side opposite 30° be 1, the side opposite 60° be √3, and the hypotenuse be 2.For angle A = 30°, the opposite side is 1 and the adjacent side is √3.Cot A = adjacent / opposite = √3 / 1 = √3.The expression to evaluate is cot A − 1/2 = √3 − 1/2.No further algebraic simplification is required; the exact value is √3 − 1/2.

Verification / Alternative check:
We can check numerically. Take √3 approximately equal to 1.732. Then cot 30° ≈ 1.732. Compute 1.732 − 0.5 = 1.232. Now evaluate each option numerically. Option a, √3 − 1/2, gives about 1.232 as expected. Option b gives about 2.232, option c gives about 0.077, option d gives about −1.232, and option e gives about 0.577. None of these except option a match the expected numeric value, confirming that √3 − 1/2 is correct.

Why Other Options Are Wrong:

• √3 + 1/2 is larger than cot 30° and does not match the expression cot A − 1/2.
• 1/√3 − 1/2 uses tan 30° rather than cot 30° and leads to a much smaller positive value.
• 1/2 − √3 is simply the negative of the correct answer.
• 1/√3 is tan 30°, not cot 30°, so it does not match the expression being evaluated.

Common Pitfalls:

• Confusing cot 30° with tan 30° and using 1/√3 instead of √3.
• Mixing up which angle is 30 degrees and which is 60 degrees in the triangle.
• Forgetting to subtract 1/2 after finding cot A and instead stopping at √3.

Final Answer:
√3 - 1/2