$f:\mathbb{R}\rightarrow\mathbb{R}$ be a twice differentiable function. Suppose $f(0)=0$. Prove that there exists some $\xi\in(-\frac{\pi}{2},\frac{\pi}{2})$ such that $$f''(\xi)=f(\xi)(1+2\tan^{2}{\xi}).$$
Please give some hint for the problem. I have tried by IVT using $()=(-x+2\tan)$