<p>The numerical value of \(\sec^2(\tan^{-1} 2) + \csc^2(\cot^{-1} 3)\) = ______.</p>
Step-by-Step Solution
Key Concept: Use the identity 1 + tan²θ = sec²θ and 1 + cot²θ = csc²θ directly with the given inverse trigonometric values, avoiding the need to find the actual angles.
<p><strong>Step 1:</strong> Let α = tan⁻¹(2), so tan(α) = 2</p><p>Using the identity sec²(α) = 1 + tan²(α):</p><p>sec²(tan⁻¹ 2) = 1 + (2)² = 1 + 4 = 5</p><p><strong>Step 2:</strong> Let β = cot⁻¹(3), so cot(β) = 3</p><p>Using the identity csc²(β) = 1 + cot²(β):</p><p>csc²(cot⁻¹ 3) = 1 + (3)² = 1 + 9 = 10</p><p><strong>Step 3:</strong> Add the two results:</p><p>sec²(tan⁻¹ 2) + csc²(cot⁻¹ 3) = 5 + 10 = <strong>15</strong></p>
Correct Answer: 15