This largest solution to this gorgeous equation is the first local extremum on a function related to the Fibonacci sequence:

$$x^2 \cdot \sin \left(\frac{2\pi}{x+1} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x} \right) \right) = (x+1)^2 \cdot \sin \left(\frac{2\pi}{x} \right) \cdot \left(3+2 \cos \left(\frac{2\pi}{x+1} \right) \right)$$

This is as simplified as I could get it. The largest solution to this equation is around $x = 2.1392.$

It appears there is no closed-form solution for this; is there any way to prove if the solution is algebraic or transcendental?

P.S. Can anyone approximate this constant to more decimal places?
**ANSWERED**