If $\left(a\cos {\theta }_i,a\sin {\theta }_i\right),i=1,2,3$  represent the vertices of an equilateral triangle inscribed in a circle , then

Given points lie in the circle $x^2+y^2=a^2$  since the points are vertices of an equilateral  triangle

$\therefore$  Circumcentre(0)  and centroid of triangle are same

$\frac{a}{3}\left(\cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3\right)=0$  and $\frac{a}{3}\left(\sin {\theta }_1+\sin {\theta }_2+sin{\theta }_3\right)=0$  

$\Rightarrow \cos {\theta }_1+\cos {\theta }_2+\cos {\theta }_3=0 and \sin {\theta }_1+\sin {\theta }_2+\sin {\theta }_3=0$

$and {\left(\cos {\theta }_1+\cos {\theta }_2\right)}^{2_{ }}+{\left(\sin {\theta }_1+\sin {\theta }_2\right)}^2= {\left(-\cos {\theta }_3\right)}^2+{\left(-\sin {\theta }_3\right)}^2 =1$

$\Rightarrow 1+1+2\left(\cos \left({\theta }_1-{\theta }_2\right)\right)=1$

 $\Rightarrow \cos \left({\theta }_1-{\theta }_2\right)=-\frac{1}{2}$