The direction cosines of a line equally inclined with the co-ordinate axes are

Direction cosines describe the direction a line points in 3D space. If a line makes angles α, β, and γ with the x-axis, y-axis, and z-axis respectively, then the direction cosines are $(cos α, cos β, cos γ)$, typically written as $(l, m, n)$.

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"Equally inclined with the coordinate axes" means the line makes the same angle with all three axes.

$α = β = γ = θ$

The direction cosines become $(cos θ, cos θ, cos θ)$.

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Direction cosines satisfy the fundamental property:

$l² + m² + n² = 1$

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Since $l = m = n = cos θ$:

$(cos θ)² + (cos θ)² + (cos θ)² = 1$

$3(cos θ)² = 1$

$cos²θ = \dfrac{1}{3}$

$cos θ = ±\dfrac{1}{\sqrt{3}}$

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The $±$ sign appears because $cos θ$ can be positive or negative depending on whether the angle is acute or obtuse. Each direction cosine can independently be $+\dfrac{1}{\sqrt{3}}$ or $-\dfrac{1}{\sqrt{3}}$.

Therefore, the direction cosines are $\left(±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}, ±\dfrac{1}{\sqrt{3}}\right)$.