<p>The numerical value of \(\cos[\tan^{-1}(-3) + \cot^{-1}(-3)]\) = ______.</p>
Step-by-Step Solution
Key Concept: Use the property that tan⁻¹(x) + cot⁻¹(x) = π/2 for x > 0, but for negative x, we must carefully apply the identity: tan⁻¹(-x) + cot⁻¹(-x) = -π/2. Then cos(-π/2) = 0.
<p><strong>Step 1:</strong> Recall the fundamental identity for inverse functions. For any real x ≠ 0:</p><p>tan⁻¹(x) + cot⁻¹(x) = π/2 when x > 0</p><p>tan⁻¹(x) + cot⁻¹(x) = -π/2 when x < 0</p><p><strong>Step 2:</strong> Since x = -3 < 0, we apply the second formula:</p><p>tan⁻¹(-3) + cot⁻¹(-3) = -π/2</p><p><strong>Step 3:</strong> Now substitute into the original expression:</p><p>cos[tan⁻¹(-3) + cot⁻¹(-3)] = cos(-π/2)</p><p><strong>Step 4:</strong> Evaluate cos(-π/2):</p><p>cos(-π/2) = cos(π/2) = 0 (since cosine is an even function)</p><p>∴ Answer: <strong>0</strong></p>
Correct Answer: 0