Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$. Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ . (Express as a fraction in lowest terms)
Let $A$ be the rotation matrix that rotates counterclockwise by angle $\theta$ about the origin, and let $B$ be the reflection matrix with the $x$-axis as the axis of reflection (axis of symmetry). Let $A = \left[\begin{array}{ll} a_{1} & a_{2} \\ a_{3} & a_{4} \end{array}\right]$ and $BA = \left[\begin{array}{ll} c_{1} & c_{2} \\ c_{3} & c_{4} \end{array}\right]$.
Given that $a_{1} + a_{2} + a_{3} + a_{4} = 2(c_{1} + c_{2} + c_{3} + c_{4})$, then $\tan\theta =$ \underline{\hspace{2cm}}. (Express as a fraction in lowest terms)