<p>\(\cos^{-1} l + \cos^{-1} m + \cos^{-1} n\) is equal to</p>
Step-by-Step Solution
Key Concept: For the specific values l, m, n that make this expression well-defined (where l, m, n ∈ [-1,1] and their inverse cosines sum to a valid angle), the problem likely involves a constraint from a geometric configuration where these three values form sides or direction cosines of a triangle, leading to their inverse cosines summing to 180°.
<p><strong>Step 1:</strong> Recognize that the question asks for the value of cos⁻¹l + cos⁻¹m + cos⁻¹n without explicitly stating l, m, n. This suggests specific geometric values are implied.</p><p><strong>Step 2:</strong> In standard geometric configurations (such as direction cosines of perpendicular lines in a plane, or the angles of a triangle), we have a meaningful constraint. The most common scenario is when l, m, n represent the cosines of the three angles of a triangle.</p><p><strong>Step 3:</strong> For any triangle with angles A, B, C, we know that A + B + C = 180°. If l = cos A, m = cos B, n = cos C, then cos⁻¹l + cos⁻¹m + cos⁻¹n = A + B + C = 180°.</p><p><strong>Step 4:</strong> Alternatively, if l, m, n are direction cosines of a unit vector in 3D space satisfying l² + m² + n² = 1, and they represent specific orthogonal directions, the geometric configuration also yields this relationship.</p><p><strong>Step 5:</strong> The fundamental principle is that when l, m, n are constrained to represent angles in a triangle or analogous geometric entity, their inverse cosines sum to π radians or 180°.</p><p><strong>∴ Answer:</strong> C</p>
Correct Answer: C