<p>Find the value of \(\cot^{-1}\left[\cot\left(\dfrac{\pi}{12} + \dfrac{\pi}{6} + \dfrac{\pi}{4}\right)\right]\) in degrees.</p>
Step-by-Step Solution
Key Concept: First simplify the argument inside cot by adding the angles, then recognize that cot⁻¹(cot(x)) = x only when x ∈ (0, π). Here the simplified angle lies outside this range, requiring angle reduction to the principal domain.
<p><strong>Step 1:</strong> Add the angles inside cot:</p><p>π/12 + π/6 + π/4 = π/12 + 2π/12 + 3π/12 = 6π/12 = π/2</p><p><strong>Step 2:</strong> Evaluate cot(π/2):</p><p>cot(π/2) = 0</p><p><strong>Step 3:</strong> Apply inverse cotangent:</p><p>cot⁻¹(0) = π/2 (since π/2 ∈ (0, π), the principal range of cot⁻¹)</p><p><strong>Step 4:</strong> Convert to degrees:</p><p>π/2 radians = (π/2) × (180/π)° = 90°</p><p>∴ Answer: <strong>90</strong></p>
Correct Answer: 90