Given \(x = \sin^{-1}\left(\frac{2}{5}\right)\), we have \(\sin x = \frac{2}{5}\).

(i) To find \(\cos^2 x\), use the identity \(\cos^2 x = 1 - \sin^2 x\).

\(\sin^2 x = \left(\frac{2}{5}\right)^2 = \frac{4}{25}\).

Thus, \(\cos^2 x = 1 - \frac{4}{25} = \frac{21}{25}\).

(ii) To find \(\tan^2 x\), use the identity \(\tan^2 x = \frac{\sin^2 x}{\cos^2 x}\).

\(\tan^2 x = \frac{\frac{4}{25}}{\frac{21}{25}} = \frac{4}{21}\).