A hyperbola, having the transverse axis of length $2 \sin \theta$, is confocal with the ellipse $3x^2+4y^2=12$ Then its equation is 

The length of transverse axis $=2\sin \theta =2a$

$\Rightarrow a=\sin \theta$

Also for ellipse $3x^2+4y^2=12$

or $\frac{x^2}{4}+\frac{y^2}{3}=1,a^2=4,b^2=3$

$e=\sqrt{1-\frac{b^2}{a^2}}=\sqrt{1-\frac{3}{4}}=\frac{1}{2}$

$\therefore$ Focus of ellipse $=\left(2\times \frac{1}{2},0\right)\Rightarrow (1,0)$

As hyperbola is confocal with ellipse, focus of hyperbola $= (1, 0)$

 $\begin{array}{l}\Rightarrow ae=1\Rightarrow \sin \theta \times e=1\\ \Rightarrow e=\mathrm{cosec}\theta \\ \therefore b^2=a^2\left(e^2-1\right)={\sin }^2\theta \left({\mathrm{cosec}}^2\theta -1\right)={\cos }^2\theta \end{array}$

$\therefore$ Equation of hyperbola is 

$\frac{x^2}{{\sin }^2\theta }-\frac{y^2}{{\cos }^2\theta }=1$

or, $x^2{\mathrm{cosec}}^2\theta -y^2{\sec }^2\theta =1$