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# A line makes angle $\theta$ with $x\:\:and\:\:z\:\:axis.$ If it makes angle $\beta$ with $y\; axis$, where $sin^2 \beta=3sin^2\theta$, then $cos^2\theta=?$

$(A)\:\:\large\frac{2}{3}\:\:\:\:\qquad\:\:(B)\:\:\frac{1}{5}\:\:\:\:\qquad\:\:(C)\:\:\frac{3}{5}\:\:\:\:\qquad\:\:(D)\:\:\frac{2}{5}.$

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• If the angle made by a line with coordinate axis are $\theta_1,\theta_2,\theta_3$, then $cos^2\theta_1+cos^2\theta_2+cos^2 \theta_3=1$
Given that the line makes angle $\theta$ with $x\:\:and\:\:z\:\:axis$ and $\beta$ with $y-axis$.
$\therefore\: co^2\theta+cos^2\theta+cos^2\beta=1$ and
$\Rightarrow\:2cos^2\theta=1-cos^2\beta$......(i)
also given that $sin^2\beta=3sin^2\theta$
$\Rightarrow\:1-cos^2\beta=3-3cos^2\theta$
$\Rightarrow\:3cos^2\theta=2+cos^2\beta$......(ii)
From (i) and (ii) $1-2cos^2\theta=3cos^2\theta-2$
$\therefore\:cos^2\theta=\large\frac{3}{5}$
answered Dec 28, 2013
edited Sep 29, 2014