How do I correctly insert a curly bracket in a table cell

Question

So I made a table in which I would like to have a curly bracket, containing three formulas in one cell. I managed to make one that look right in the pdf, tho I still get an error message.

So this is what I want, and also what I get when I export to pfd. But there are a few errors that come up which I don't know how to fix.

    \begin{center}
        \begin{tabular}{l l}
            \hline
            Polaire vergelijking & Cartesiaans equivalent\\
            \hline
            $r \cos\theta = 2$ & $x=2$\\
            $r^2\cos \theta \sin \theta = 4$ & $xy=4$\\
            $r^2\cos^2\theta-r^2\sin^2\theta=1$ & $x^2-y^2=1$\\
            $r=1+2r\cos\theta$ & \left\{ \begin{tabular} {@{}l@{}} $y^2-3x^2-4x-1=0$\\  $\sqrt{x^2+y^2}=1+2x$\\ $x^2+y^2=(1+2x)^2$ \end{tabular} \right.\\
            \hline
        \end{tabular}
    \end{center}

Answer 1

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Instead nested tabular is used cases environment defined in the amsmath package. For more vertical space is added makecell package to preamble and in code for table used its macro \makegapedcells. Instead tabular is used array and it is enclosed in math environment. By this you not need to write $ before and after each equations in table:

\documentclass{article}
\usepackage{amsmath}  % <---
\usepackage{makecell} % <---

\begin{document}
\[    
\setcellgapes{3pt} % <---
\makegapedcells    % <---
    \begin{array}{l l}
        \hline
    $Polaire vergelijking$ & $Cartesiaans equivalent$   \\
        \hline
r \cos\theta = 2                    & x=2           \\
r^2\cos \theta \sin \theta = 4      & xy=4          \\
r^2\cos^2\theta-r^2\sin^2\theta=1   & x^2-y^2=1     \\
r=1+2r\cos\theta                    & \begin{cases} 
                            y^2-3x^2-4x-1=0         \\  
                            \sqrt{x^2+y^2}=1+2x     \\ 
                            x^2+y^2=(1+2x)^2    
                                      \end{cases}   \\
        \hline
    \end{array}
\]
\end{document}