<p>If direction cosines of a line are \(\langle l, m, n \rangle\) such that \(l^2 + m^2 + n^2 = 1\) and \(\alpha = \theta\), \(\beta = \beta\), \(\gamma = \theta\), then \(\cos^2\theta\) equals:</p>
Step-by-Step Solution
Key Concept: Direction cosines satisfy the fundamental identity l² + m² + n² = 1. When angles α, β, γ are the angles a line makes with coordinate axes, their cosines are the direction cosines themselves: l = cos α, m = cos β, n = cos γ.
Step 1: Identify that l, m, n are direction cosines, so l = cos α, m = cos β, n = cos γ. Step 2: Given that α = θ, β = θ, γ = θ, we have: l = cos θ, m = cos θ, n = cos θ Step 3: Apply the fundamental identity l^2 + m^2 + n^2 = 1: cos^2θ + cos^2θ + cos^2θ = 1 3cos^2θ = 1 Step 4: Solve for cos^2θ: cos^2θ = 1/3 ∴ Answer: C
Correct Answer: C