(i) Start with the equation \(2 \cos x = 3 \tan x\). Replace \(\tan x\) with \(\frac{\sin x}{\cos x}\), giving \(2 \cos x = 3 \frac{\sin x}{\cos x}\).

Multiply through by \(\cos x\) to eliminate the fraction: \(2 \cos^2 x = 3 \sin x\).

Use the identity \(\cos^2 x = 1 - \sin^2 x\) to rewrite the equation: \(2(1 - \sin^2 x) = 3 \sin x\).

Expand and rearrange: \(2 - 2 \sin^2 x = 3 \sin x\).

Rearrange to form a quadratic equation: \(2 \sin^2 x + 3 \sin x - 2 = 0\).

(ii) Start with the equation \(2 \cos 2y = 3 \tan 2y\). Replace \(\tan 2y\) with \(\frac{\sin 2y}{\cos 2y}\), giving \(2 \cos 2y = 3 \frac{\sin 2y}{\cos 2y}\).

Multiply through by \(\cos 2y\) to eliminate the fraction: \(2 \cos^2 2y = 3 \sin 2y\).

Use the identity \(\cos^2 2y = 1 - \sin^2 2y\) to rewrite the equation: \(2(1 - \sin^2 2y) = 3 \sin 2y\).

Expand and rearrange: \(2 - 2 \sin^2 2y = 3 \sin 2y\).

Rearrange to form a quadratic equation: \(2 \sin^2 2y + 3 \sin 2y - 2 = 0\).

Solve the quadratic equation for \(\sin 2y\): \(\sin 2y = \frac{-3 \pm \sqrt{3^2 - 4 \cdot 2 \cdot (-2)}}{2 \cdot 2}\).

Calculate the discriminant: \(9 + 16 = 25\).

\(\sin 2y = \frac{-3 \pm 5}{4}\).

\(\sin 2y = \frac{2}{4} = \frac{1}{2}\) or \(\sin 2y = \frac{-8}{4} = -2\) (not possible).

For \(\sin 2y = \frac{1}{2}\), \(2y = 30^\circ\) or \(2y = 150^\circ\).

Thus, \(y = 15^\circ\) or \(y = 75^\circ\).