Why there is no closed form solution to $\tan(x) + x = \frac{\pi}{2}$

trigonometry

I want to solve for x…

$$\tan(x) + x = \frac{\pi}{2}$$
where $0 < x < 1$

…but i can't.

Someone wrote to me saying that this have no closed form solution (very sad).

It's really true?
No closed form solution?
Maybe sine or cosine of a rational or a rational part of $\pi$

EDIT: Approximation
$$x \approx 0.710462737775516856747428268$$

EDIT: I've tried to solve this equation:

$$\tan(\tan(x)) = \frac{1}{\tan(x)}$$

I suppose they have the same solution.

Best Answer

The general rule is: equations have no closed-form roots.

When the equations are polynomials of degree no exceeding $4$, the roots have always a closed-form. But for higher degrees, this is exceptional.

For transcendental equations that cannot be reduced to algebraic ones, AFAIK we have no general resolution method (not even to just tell the number of roots).

Related Question