Question #41885

Let $L_1$ and $L_2$ be two lines in 3D space with direction ratios $\langle a_1, b_1, c_1 \rangle$ and $\langle a_2, b_2, c_2 \rangle$ respectively, where: $a_1 = \lim_{x \to 0} \frac{6(\sin x - x)}{x^3}, \quad b_1 = \lim_{x \to 0} \frac{x^2}{\ln(\sec x)}, \quad c_1 = \lim_{x \to 0} \frac{6(\tan x - x)}{x^3}$ and $a_2 = \lim_{x \to 0} \frac{2(1 - \cos^3(2x))}{3x^2}, \quad b_2 = \lim_{x \to 0} \frac{16(\sqrt{\cos x} - 1)}{x^2}, \quad c_2 = \lim_{x \to 0} \frac{21(\sin(2x) - 2x)}{4x^3}$ If $\theta$ is the angle between the lines $L_1$ and $L_2$, then the value of $\cos \theta$ is: