CUET UG 2025 — Mathematics Calculus

When a line makes angles $\alpha$α, $\beta$β, and $\gamma$γ with the positive directions of the x, y, and z axes, the values $\cos\alpha$cosα, $\cos\beta$cosβ, and $\cos\gamma$cosγ are called direction cosines.

For any line in 3D space:

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

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The double angle formula states:

$$\cos 2\theta = 2\cos^2\theta - 1$$cos2θ=2cos2θ−1

Applying this to each angle:

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

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

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

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Adding the three expressions:

$$\cos 2\alpha + \cos 2\beta + \cos 2\gamma$$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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Using the property of direction cosines:

$$= 2(1) - 3$$=2(1)−3

$$= -1$$=−1

Therefore, $\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1$cos2α+cos2β+cos2γ=−1