If θ is an acute angle and 4 sin²θ = 3, then what is the exact value of the expression (tan θ − cot θ) / 2?

Introduction / Context:
This question combines trigonometric identities with algebraic manipulation. You are given a condition on sin²θ and asked to evaluate a compound expression involving tan θ and cot θ. Such questions check conceptual clarity about basic identities and careful handling of square roots for acute angles.

Given Data / Assumptions:

• 4 sin²θ = 3.
• θ is acute, so 0° < θ < 90° and all primary trigonometric ratios are positive.
• We need to find (tan θ − cot θ)/2.

Concept / Approach:
Key concepts:

• From 4 sin²θ = 3, we can find sin θ and then cos θ using sin²θ + cos²θ = 1.
• Once sin θ and cos θ are known, compute tan θ = sin θ / cos θ and cot θ = cos θ / sin θ.
• Simplify tan θ − cot θ and then divide by 2, making sure to keep everything in exact radical form.

Step-by-Step Solution:
Step 1: From 4 sin²θ = 3, divide both sides by 4 to get sin²θ = 3/4. Step 2: Since θ is acute, sin θ is positive, so sin θ = √(3/4) = √3 / 2. Step 3: Use sin²θ + cos²θ = 1 to find cos θ: cos²θ = 1 − 3/4 = 1/4, hence cos θ = 1/2 for an acute angle. Step 4: Compute tan θ = sin θ / cos θ = (√3 / 2) / (1/2) = √3. Step 5: Compute cot θ = cos θ / sin θ = (1/2) / (√3 / 2) = 1 / √3. Step 6: Now tan θ − cot θ = √3 − 1/√3 = (3 − 1)/√3 = 2/√3. Step 7: Finally, (tan θ − cot θ)/2 = (2/√3)/2 = 1/√3.
Verification / Alternative check:
Step 1: Notice that sin θ = √3 / 2 corresponds to θ = 60° in standard trigonometry. Step 2: At θ = 60°, tan θ = √3 and cot θ = 1 / √3, which matches our derived values. Step 3: Substitute θ = 60° directly into the target expression: (tan 60° − cot 60°)/2 = (√3 − 1/√3)/2 = (2/√3)/2 = 1/√3, confirming the result.
Why Other Options Are Wrong:
Option 1 corresponds to tan θ − cot θ = 2, which is not supported by the computed exact values. Option 0 would imply tan θ = cot θ, which cannot happen for this angle since their values are different. Option √3 is the value of tan θ alone, not the simplified expression asked in the problem. Option 2/√3 is the value of tan θ − cot θ before dividing by 2, so it is only an intermediate result, not the final answer.
Common Pitfalls:
Some learners forget that θ is acute and may take a negative square root for sin θ or cos θ. Another typical error is to stop at tan θ − cot θ without applying the final division by 2. Careless algebra with radicals, such as mishandling √3 − 1/√3, often leads to incorrect simplification.
Final Answer:
The exact value of (tan θ − cot θ)/2 is 1/√3.