From any point on the circle $x^2+y^2=a^2$ tangents are drawn to the circle $x^2+y^2=a^2{\sin }^2\theta$. The angle between them is

A point on $x^2+y^2=a^2\text{ is }P(a\cos \theta ,a\sin \theta )$

$\text{ Radius of }S\equiv x^2+y^2-a^2{\sin }^2\theta =0\text{ is }r=a\sin \theta$

Length of the tangent from $P\text{ to }S=0\text{ is }\sqrt{S_{11}}=\sqrt{a^2{\cos }^2\theta +a^2{\sin }^2\theta -a^2{\sin }^2\theta }$

$\tan \frac{\alpha }{2}=\frac{r}{\sqrt{S_{11}}}=\frac{a\sin \theta }{a\cos \theta }=\tan \theta \Rightarrow \frac{\alpha }{2}=\theta \Rightarrow \alpha =2\theta$