Weird "hidden answer" in $2\tan(2x)=3\text{cot}(x)$ The question is

Find the solutions to the equation $$2\tan(2x)=3\cot(x) , \space 0<x<180$$

I started by applying the tan double angle formula and recipricoal identity for cot
$$2* \frac{2\tan(x)}{1-\tan^2(x)}=\frac{3}{\tan(x)}$$
$$\implies 7\tan^2(x)=3 \therefore x=\tan^{-1}\left(-\sqrt\frac{3}{7} \right)$$
$$x=-33.2,33.2$$
Then by using the quadrants

I was lead to the final solution that $x=33.2,146.8$ however the answer in the book has an additional solution of $x=90$, I understand the reasoning that $\tan(180)=0$ and $\cot(x)$ tends to zero as x tends to 90 however how was this solution found?
Is there a process for consistently finding these "hidden answers"?

The error is in your very first line.
As alluded to by @Tavish in the comments, since the double-angle identity for $\tan$ has domain $$\mathbb R\setminus \left\{(2k+1)\frac{\pi}2 \mid k\in\mathbb Z\right\},$$ a separate case for odd multiples of $90^{\circ}$ ought to be created when applying it. (In doing so while solving an equation where the flow of logic is $“\implies”$ rather than $“\iff”,$ we are merely asserting that this separate case contains potential solutions.)

*

*So, for $x\in\left(0^{\circ},180^{\circ}\right),$
$$2\tan(2x)=3\cot(x) \tag1$$ $$\implies
   \frac{4\tan(x)}{1-\tan^2(x)}=\frac{3}{\tan(x)} \quad\text{or}\quad
   x=90^{\circ} \\ \implies \tan^2(x)=\frac37 \quad\text{or}\quad
   x=90^{\circ}$$ $$\implies x=33.2^{\circ}\:\text{or}\:146.8^{\circ}
   \quad\text{or}\quad x=90^{\circ}; \tag2 $$ plugging (2) into (1)
reveals that the full solution set is
$\left\{33.2^{\circ},90^{\circ},146.8^{\circ}\right\}.$


*Alternatively, using “$\iff$”:
for $x\in\left(0^{\circ},180^{\circ}\right),$
$$2\tan(2x)=3\cot(x) \\ \iff
   \left[\frac{4\tan(x)}{1-\tan^2(x)}=\frac{3}{\tan(x)}
   \quad\text{or}\quad x=90^{\circ}\right] \quad\text{and}\quad
   \cos(2x),\sin(x)\neq0\\ \iff \left[\tan^2(x)=\frac37
   \quad\text{or}\quad x=90^{\circ}\right] \quad\text{and}\quad
   \cos(2x),\sin(x)\neq0\\$$ $$\iff x=33.2^{\circ}\:\text{or}\:
   90^{\circ}\:\text{or}\: 146.8^{\circ}.$$
