If a line makes angles \(\alpha, \beta\) and \(\gamma\) with positive x-axis, y-axis and z-axis respectively, then the value of \(\sin^2\displaystyle\frac{\alpha}{2}.\cos^2\displaystyle\frac{\alpha}{2}+\sin^2\displaystyle\frac{\beta}{2}.\cos^2\displaystyle\frac{\beta}{2}+ \sin^2\displaystyle\frac{\gamma}{2}. \cos^2\displaystyle\frac{\gamma}{2}\) is
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\displaystyle\frac{1}{2}\)
Since \(\sin\theta\cos\theta = \frac{1}{2}\sin 2\theta\), each term simplifies:
\(\displaystyle \sin^2\frac{\alpha}{2}\cos^2\frac{\alpha}{2} = \left(\frac{1}{2}\sin\alpha\right)^2 = \frac{\sin^2\alpha}{4}\)
So the expression is
\(\displaystyle \frac{\sin^2\alpha + \sin^2\beta + \sin^2\gamma}{4}\)
The direction cosines \(\cos\alpha, \cos\beta, \cos\gamma\) satisfy \(\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1\), so
\(\sin^2\alpha + \sin^2\beta + \sin^2\gamma = 3 - (\cos^2\alpha + \cos^2\beta + \cos^2\gamma)\)
\(= 3 - 1 = 2\)
The required value is \(\frac{2}{4} = \frac{1}{2}\).