If \(x=f(y)\) is the solution of the differential equation \(\left(1+y^2\right)+\left(x-2 e^{\tan ^{-1} y}\right) \frac{…

If \(x=f(y)\) is the solution of the differential equation

\(\left(1+y^2\right)+\left(x-2 e^{\tan ^{-1} y}\right) \frac{d y}{d x}=0, y \in\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)\)

with \(f(0)=1\), then \(f\left(\frac{1}{\sqrt{3}}\right)\) is equal to :

[JEE Main 2025, 22 Jan (Shift 2)]