In this case it might be simplest to do that by hand. (I also replaced grey by gray.)
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=1.3]
\draw [->, ultra thick, gray](0,-6)--(0,6) node[right,font=\fontsize {30}
{30}\selectfont]{$x$};
\draw [->,ultra thick, gray](-6,0)--(6,0) node[above,font=\fontsize {30}
{30}\selectfont]{$y$};
\draw (6,-2)--(-6,6);
\draw (-4,-4)--(2,5);
\draw (0.3,{2-0.2}) -- ++(-0.2,-0.3)--++(-0.3,0.2);
\draw[font=\fontsize{15}{15}\selectfont,black]
--(3,1) node {$\tan\theta_2$};
\fill[yellow] (3,0)-- ++(4mm,0mm)
arc[start angle=0, end angle=143.130102, radius=4mm]--cycle;
\fill[yellow] (-1.333333333,0)-- ++(4mm,0mm)
arc[start angle=0, end angle=56.3099, radius=4mm]--cycle;
\draw[font=\fontsize{10}{10}\selectfont, black]
--(-.5,.4) node {$\tan\theta_1$};
\end{tikzpicture}
\end{document}

EDIT: Here comes a perhaps even simpler alternative with explanation.
\documentclass{article}
\usepackage{tikz}
\begin{document}
\begin{tikzpicture}[scale=1.3]
\draw [->, ultra thick, gray](0,-6)--(0,6) node[right,font=\fontsize {30}
{30}\selectfont]{$x$};
\draw [->,ultra thick, gray](-6,0)--(6,0) node[above,font=\fontsize {30}
{30}\selectfont]{$y$};
\draw (6,-2)--(-6,6);
\draw (-4,-4)--(2,5);
% the slope of the first line is -12/8, so this defines the rotation angle
\draw[rotate around={atan2(8,-12):(0,2)}] (0,2) rectangle ++(-0.3,0.3);
%\draw (0.3,{2-0.2}) -- ++(-0.2,-0.3)--++(-0.3,0.2);
\draw[font=\fontsize{15}{15}\selectfont,black]
--(3,1) node {$\tan\theta_2$};
\fill[yellow] (3,0)-- ++(4mm,0mm)
arc[start angle=0, end angle=143.130102, radius=4mm]--cycle;
\fill[yellow] (-1.333333333,0)-- ++(4mm,0mm)
arc[start angle=0, end angle=56.3099, radius=4mm]--cycle;
\draw[font=\fontsize{10}{10}\selectfont, black]
--(-.5,.4) node {$\tan\theta_1$};
\end{tikzpicture}
\end{document}