Question

If \(\alpha, \beta, \gamma\) are angles of inclinations of a line with \(x, y\) and \(z\)-axis respectively, then which of the following are correct?
A. \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\)
B. \(\cos 2\alpha + \cos 2\beta + \cos 2\gamma = -1\)
C. \(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = -2\)
D. \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 2\)
Choose the correct answer from the options given below:

• A. A and B only
• B. A, B and C only
• C. B and D only
• D. A and D only

Hint:

Use \(l^2+m^2+n^2=1\) for direction cosines, then \(\cos 2\theta = 2\cos^2\theta-1\).

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
The direction cosines of a line are \(l=\cos\alpha\), \(m=\cos\beta\) and \(n=\cos\gamma\). Here \(\alpha, \beta, \gamma\) are the angles the line makes with the x, y and z axes.

Step 2: Key Formula or Approach:
For any line in space, the direction cosines satisfy \(l^2+m^2+n^2=1\). This is because \((l,m,n)\) is a unit vector along the line. We also use \(\cos 2\theta = 2\cos^2\theta - 1\) and \(\sin^2\theta = 1-\cos^2\theta\).

Step 3: Check statement (A).
By the direction cosine identity, \(\cos^2\alpha+\cos^2\beta+\cos^2\gamma = 1\). So (A) is TRUE.

Step 4: Check statement (B).
Use \(\cos 2\theta = 2\cos^2\theta-1\) on each angle and add:
\[ \cos 2\alpha+\cos 2\beta+\cos 2\gamma = 2(\cos^2\alpha+\cos^2\beta+\cos^2\gamma) - 3 = 2(1)-3 = -1 \]
So (B) is TRUE.

Step 5: Check statement (C).
Use \(\sin^2\theta = 1-\cos^2\theta\):
\[ \sin^2\alpha+\sin^2\beta+\sin^2\gamma = 3 - 1 = 2 \]
The statement says \(-2\). A sum of squares can never be negative, so (C) is FALSE.

Step 6: Check statement (D).
The sum of the squares of the cosines is \(1\), not \(2\). So (D) is FALSE.

Step 7: Match with the options.
Only A and B are correct. This is option 1. Option 2 wrongly includes C. Options 3 and 4 wrongly include D.

Final Answer:
Statements A and B are correct, which is option 1. \[ \boxed{\text{A and B only (Option 1)}} \]