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If $\alpha ,\beta ,\gamma $ be the angles made by a line with $x, y$ and $z$ axes respectively so that $2\left( {\frac{{{{\tan }^2}\,\alpha }}{{1 + {{\tan }^2}\,\alpha }} + \frac{{{{\tan }^2}\,\beta }}{{1 + {{\tan }^2}\,\beta }} + \frac{{{{\tan }^2}\,\gamma }}{{1 + {{\tan }^2}\,\gamma }}} \right) = 3\,{\sec ^2}\,\frac{\theta }{2},$ then $\theta =$
$\frac{\pi }{{12}}$
$\frac{\pi }{{10}}$
$\frac{\pi }{{6}}$
$\frac{\pi }{{3}}$
Solution
We know that.
$\cos ^{2} \alpha+\cos ^{2} \beta+\cos ^{2} \gamma=1 \Rightarrow \Sigma \cos ^{2} \alpha=1 \ldots(i)$
Given that
$2\left(\frac{\tan ^{2} \alpha}{1+\tan ^{2} \alpha}+\frac{\tan ^{2} \beta}{1+\tan ^{2} \beta}+\frac{\tan ^{2} \gamma}{1+\tan ^{2} \gamma}\right)=3 \sec ^{2} \frac{\theta}{2}$
${\Rightarrow 2\left(\sin ^{2} \alpha+\sin ^{2} \beta+\sin ^{2} \gamma\right)=3 \sec ^{2} \frac{\theta}{2}} $
${\Rightarrow 4 \cos ^{2} \frac{\theta}{2}=3 \quad[\text { From }(\mathrm{i})]} $
${\Rightarrow \cos \theta=\frac{1}{2} \Rightarrow \theta=\frac{\pi}{3}}$
