CUET UG 2025 — Mathematics Geometry

When a line makes angles α, β, γ with the x-axis, y-axis, and z-axis, the direction cosines are:

$$l = \cos \alpha$$l=cosα

$$m = \cos \beta$$m=cosβ

$$n = \cos \gamma$$n=cosγ

The fundamental property of direction cosines:

$$\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$$cos2α+cos2β+cos2γ=1

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Using the double angle formula $\cos2\theta = 2\cos^2\theta - 1$cos2θ=2cos2θ−1 for each angle:

$$\cos2\alpha = 2\cos^2\alpha - 1$$cos2α=2cos2α−1

$$\cos2\beta = 2\cos^2\beta - 1$$cos2β=2cos2β−1

$$\cos2\gamma = 2\cos^2\gamma - 1$$cos2γ=2cos2γ−1

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Adding these expressions:

$$\cos2\alpha + \cos2\beta + \cos2\gamma$$cos2α+cos2β+cos2γ

$$= (2\cos^2\alpha - 1) + (2\cos^2\beta - 1) + (2\cos^2\gamma - 1)$$=(2cos2α−1)+(2cos2β−1)+(2cos2γ−1)

$$= 2\cos^2\alpha + 2\cos^2\beta + 2\cos^2\gamma - 3$$=2cos2α+2cos2β+2cos2γ−3

$$= 2(\cos^2\alpha + \cos^2\beta + \cos^2\gamma) - 3$$=2(cos2α+cos2β+cos2γ)−3

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Substituting $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$cos2α+cos2β+cos2γ=1:

$$= 2(1) - 3$$=2(1)−3

$$= -1$$=−1

Therefore, $\cos2\alpha + \cos2\beta + \cos2\gamma = -1$cos2α+cos2β+cos2γ=−1