What is the value of $\tan\left(\cos^{-1} \frac{5}{13}\right)$?

Explanation

1. 01

   Let $\theta = \cos^{-1} \frac{5}{13}$: adjacent 5, hypotenuse 13, so opposite $\sqrt{169 - 25} = 12$.

2. 02

   $\tan \theta = \frac{12}{5}$, positive since $\theta$ is acute.