Evaluate $\;$ $\cos \left(\sin^{-1} \dfrac{5}{13}\right)$
Let $\;$ $\sin^{-1} \left(\dfrac{5}{13}\right) = \theta$ $\;\;\; \cdots \; (1)$
Then, $\;$ $\sin \theta = \dfrac{5}{13}$
and $\cos \theta = \sqrt{1 - \sin^2 \theta} = \sqrt{1 - \dfrac{25}{169}} = \sqrt{\dfrac{144}{169}} = \dfrac{12}{13}$ $\;\;\; \cdots \; (2)$
Now, $\;$ $\cos \left(\sin^{-1} \dfrac{5}{13}\right) = \cos \theta = \dfrac{12}{13}$ $\;\;\;$ [By equations $(1)$ and $(2)$]