Question

If a line in the space makes angles \(\alpha,\beta,\gamma\) with the coordinate axes, then
\[ \cos2\alpha+\cos2\beta+\cos2\gamma+\sin^2\alpha+\sin^2\beta+\sin^2\gamma \] equals

• A. \(-1\)
• B. \(0\)
• C. \(1\)
• D. \(2\)

Hint:

Always remember: direction cosines satisfy \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1\). Convert the given expression into this form.

Answer 1

The Correct Option is C

Step 1: Use identity \(\cos2\theta = 1-2\sin^2\theta\).
So:
\[ \cos2\alpha+\sin^2\alpha = (1-2\sin^2\alpha)+\sin^2\alpha=1-\sin^2\alpha=\cos^2\alpha \]
Similarly:
\[ \cos2\beta+\sin^2\beta=\cos^2\beta \]
\[ \cos2\gamma+\sin^2\gamma=\cos^2\gamma \]
Step 2: Add all three results.
\[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma \]
Step 3: Use direction cosine identity.
For any line in space:
\[ \cos^2\alpha+\cos^2\beta+\cos^2\gamma=1 \]
Final Answer:
\[ \boxed{1} \]