(i) Start with the identity \(\cos^2 x = 1 - \sin^2 x\).

Then, \(\cos^4 x = (\cos^2 x)^2 = (1 - \sin^2 x)^2\).

Expanding, \((1 - \sin^2 x)^2 = 1 - 2\sin^2 x + \sin^4 x\).

Thus, \(\cos^4 x = 1 - 2\sin^2 x + \sin^4 x\).

(ii) Substitute \(\cos^4 x = 1 - 2\sin^2 x + \sin^4 x\) into the equation:

\(8\sin^4 x + (1 - 2\sin^2 x + \sin^4 x) = 2(1 - \sin^2 x)\).

Simplify to get \(9\sin^4 x + 1 - 2\sin^2 x = 2 - 2\sin^2 x\).

Thus, \(9\sin^4 x = 1\).

\(\sin^4 x = \frac{1}{9}\).

\(\sin^2 x = \frac{1}{3}\).

\(\sin x = \pm \sqrt{\frac{1}{3}}\).

Solving for \(x\) gives \(x = 35.3^\circ, 144.7^\circ, 215.3^\circ, 324.7^\circ\).