If a line makes angles α, β, γ with the positive x, y and z-axes, prove that cos²α + cos²β + cos²γ = 1. Hence find the direction cosines of a line that makes equal angles with the axes.

1. 1 · Write what is given and what is to be found. Given: a line through the origin making angles α, β, γ with the axes. To prove: cos²α + cos²β + cos²γ = 1.
2. 2 · Draw the figure or sketch. Draw the x, y and z-axes from O. Draw OP along the line with P(x, y, z) and OP = r. Mark α between OP and the x-axis.
3. 3 · Write the rule. In the right triangle from P to the x-axis, cos α = x/r. So x = r cos α, y = r cos β, z = r cos γ.
4. 4 · Use the distance formula.
   • r² = x² + y² + z²
   • r² = r²(cos²α + cos²β + cos²γ)
   • Divide by r²: cos²α + cos²β + cos²γ = 1
5. 5 · Apply it to equal angles. α = β = γ gives 3 cos²α = 1, so cos α = ±1/√3.
6. 6 · Check and box the answer. Hence proved. The direction cosines are ±1/√3, ±1/√3, ±1/√3. Check: 1/3 + 1/3 + 1/3 = 1.

Scoring tip: State x = r cos α clearly. The first mark is for this projection step.