The length of the chord joining $\theta$ and $\phi$ on the circle $x^2+y^2=a^2$ is 

    Equation of the chord joining the points $\theta$ and $\varphi$ on the circle $x^2+y^2=a^2$ is

$x \cos \left(\frac{\theta +\varphi }{2}\right)+y \sin \left(\frac{\theta +\varphi }{2}\right)=a \cos \left(\frac{\theta -\varphi }{2}\right)$

d= Distance from the centre $\left(0, 0\right)$ to the chord $x \cos \left(\frac{\theta +\varphi }{2}\right)+y \sin \left(\frac{\theta +\varphi }{2}\right)=a \cos \left(\frac{\theta -\varphi }{2}\right)$$\Rightarrow d=a \cos$$\left(\frac{\theta -\varphi }{2}\right)$

Length of the chord $=2\sqrt{a^2-a^2 {\cos }^2 \left(\frac{\theta -\varphi }{2}\right)}=2a \sin \left(\frac{\theta -\varphi }{2}\right).$