Explanation

(c)
Let $PQ$ and $RP$ be the two tangents and $P$ be the point on the circle $x^{2} + y^{2} = a^{2}$ whose coordinates are $(a\cos t,\ a\sin t)$ and $\angle OPQ = \theta$
Now, $PQ =$length of tanget from $P$ on the circle
$x^{2} + y^{2} = a^{2}\sin^{2}\alpha$
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$\therefore PQ = \sqrt{a^{2}\cos^{2}t + a^{2}\sin^{2}t - a^{2}\sin^{2}\alpha}$
$= \sqrt{a^{2}\left( \cos^{2}t + \sin^{2}t \right) - a^{2}\sin^{2}\alpha}$
$= a\cos\alpha\ \ \ (\because\cos^{2}t + \sin^{2}t = 1)$
and $OQ =$radius of the circle $(x^{2} + y^{2} = a^{2}\sin^{2}\alpha)$
$\Rightarrow OQ = a\sin\alpha$
$\therefore\tan\theta = \frac{OQ}{PQ} = \frac{a\sin\alpha}{a\cos\alpha} = \tan\alpha\ \ \Rightarrow \ \ \theta = \alpha$
$\therefore$ Angle between tangents $= \angle QPR = 2\theta = 2\alpha$