Question

If a line in the space makes angle \( \alpha, \beta, \gamma \) with the coordinate axes, then \( \cos 2\alpha + \cos 2\beta + \cos 2\gamma + \sin^{2}\alpha + \sin^{2}\beta + \sin^{2}\gamma \) equals

• A. -1
• B. 0
• C. 1
• D. 2

Hint:

Sum of squares of direction cosines ($l^2+m^2+n^2$) is always 1.

Answer 1

The Correct Option is C

Step 1: Use Double Angle Formula
$\cos 2\theta = \cos^{2} \theta - \sin^{2} \theta$. Expression $= (\cos^{2} \alpha - \sin^{2} \alpha) + (\cos^{2} \beta - \sin^{2} \beta) + (\cos^{2} \gamma - \sin^{2} \gamma) + \sin^{2} \alpha + \sin^{2} \beta + \sin^{2} \gamma$.
Step 2: Simplify
Expression $= \cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma$.
Step 3: Identity for Direction Cosines
For any line in space, $\cos^{2} \alpha + \cos^{2} \beta + \cos^{2} \gamma = 1$.
Final Answer: (c)