The angle between the tangents to the curve $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$ at the point $(a,0)$ and $(0,b)$ is

We have, $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$

$\Rightarrow \frac{2x}{a^2}+\frac{2y}{b^2}\frac{dy}{dx}=0\Rightarrow \frac{dy}{dx}=\frac{-b^{2}x}{a^{2}y}$

$\therefore {\tan {\theta }_1=\frac{dy}{dx}]}_{(a,0)}=\infty \Rightarrow {\theta }_1=\frac{\pi }{2}$ and $\therefore {\tan {\theta }_2=\frac{dy}{dx}]}_{(0,b)}=0\Rightarrow {\theta }_2=0$

Hence, the angle between the two tangents is $\theta ={\theta }_1-{\theta }_2=\frac{\pi }{2}-0=\frac{\pi }{2}$