Question

If a line makes angles \(α,β,γ\) with the coordinate axes, then the sum of values of \(sin^2α+sin^2β+sin^2γ\) and \(cos2α+cos2β+cos2γ\) is ...

• A. \(5\)
• B. \(0\)
• C. \(3\)
• D. \(1\)

Hint:

Direction cosines satisfy \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\).

Answer 1

The Correct Option is D

Step 1: Plan:
Pick a concrete line, like the X-axis, to check.

Step 2: Steps:
For the X-axis, $\alpha = 0$ and $\beta=\gamma = 90^{\circ}$. The first sum is $0+1+1 = 2$. The second is $\cos0 + \cos180^{\circ}+\cos180^{\circ} = 1 - 1 - 1 = -1$. The total is $1$, and since the result holds for every line, the answer is $1$.

Final Answer:
The required sum is $1$, option (D). \[ \boxed{1} \]