Question #41932

Let $L_1$ be the line of intersection of the planes $P_1$ and $P_2$, where $P_1: x + y + z = 0$ and $P_2$ is given by the equation $\lim_{t \to 0} \frac{1}{t} \begin{vmatrix} x & y & z \\ \sin t & \cos t & 1 \\ \sin 2t & \cos 2t & 1 \end{vmatrix} = 0$ Let $L_2$ be the line of intersection of the planes $P_3$ and $P_4$, where $P_3: x + y + 4z = 0$ and $P_4$ is given by the equation $\lim_{s \to 0} \frac{1}{s^2} \begin{vmatrix} x & y & z \\ 1 & 1 & -1 \\ e^{s^2} & \cosh(2s) & -e^{s^2} \end{vmatrix} = 0$ If $\theta$ is the acute angle between the lines $L_1$ and $L_2$, then the value of $\theta$ is: