Nonlinear functional equation with tangent

Question

Please help me with this. I need to find a non-trivial function $g(x)$ which satisfy the following functional equation $$\tan(g(x))+g(x)+g(x)\tan^2(g(x))=0$$

Answer 1

There is no non-trivial real (I assume you meant real as you didn't specify anything else) solution. We can rewrite your equation (using the usual trigonometric identities) as

$$g(x) = - \frac{\tan(g(x))}{1+\tan^2(g(x))} = -\frac{1}{2} \sin(2 g(x))$$

If there is some $x_0 \in \mathbb R$ such that $a := g(x_0) \neq 0$ then we have

$$a = - \frac{1}{2} \sin(2 a)$$

which is equivalent to $-2a = \sin(2a)$ which has the only real solution $a=0$. Therefore $g(x_0)=0$ and therefore $g(x) = 0$ for all $x$.