Answer
$\frac{2\sqrt {21}}{21}$
Work Step by Step
Step 1. For $tan(sin^{-1}\frac{2}{5})$, let $u=sin^{-1}\frac{2}{5}$, we have $sin(u)=\frac{2}{5}$
Step 2 As the range of inverse sine is within $(-\frac{\pi}{2}, \frac{\pi}{2})$, we know $0\lt u\lt \frac{\pi}{2}$
Step 3. Use the Pythagorean Identify $cos^2u=1-sin^2u=1-\frac{4}{25}=\frac{21}{25}$ which gives $cos(u)=\frac{\sqrt {21}}{5}$
Step 4. $tan(sin^{-1}\frac{2}{5})=tan(u)=\frac{sin(u)}{cos(u)}=\frac{2}{\sqrt {21}}=\frac{2\sqrt {21}}{21}$
