$\begin{aligned} & \cos \left[2 \cos ^{-1} x+\sin ^{-1} x\right] \\ & =\cos \left[\cos ^{-1} x+\cos ^{-1} x+\sin ^{-1} x\right] \\ & =\cos \left[\cos ^{-1} x+\pi / 2\right]=-\sin \left[\cos ^{-1} x\right] \\ & =-\sin \left[\sin ^{-1} \sqrt{1-x^2}\right]=-\sqrt{1-x^2} \\ & =-\sqrt{1-\left(\frac{1}{5}\right)^2}=-\sqrt{\frac{24}{25}}=-\frac{2 \sqrt{6}}{5} \\ & \end{aligned}$