For the following general trigonometric function (such that $a,x\in\mathbb{R}$):
$\displaystyle \pi x+\arctan\left(\cot\pi x\right)=a$
How would you solve for $x$? I ended up with $\cot\pi x=\tan\left(a-\pi x\right)$ and $x=\frac{\cot^{-1}\left(\tan\left(a-\pi x\right)\right)+\pi m}{\pi}$. However, I can't isolate $x$ on the LHS. I would be grateful for any help. For similar situations, with polynomials, I have used the Lambert W function. Is there an equivalent for trigonometric functions.
Edit: I have approximated the above to
$\displaystyle \pi x+\frac{8\pi\cot\left(\pi x\right)}{3\pi+\sqrt{25\pi^{2}+256\cot^{2}\left(\pi x\right)}}=a$
I don't know how to convert the cotangent term to polynomial form (which would allow simplification) and then $x$ can (hopefully) be isolated. A taylor series might be too complicated?