(a) Differentiate \(y = \frac{x}{\cos^2 x}\) using the quotient rule:

\(\frac{dy}{dx} = \frac{\cos^2 x + 2x \sin x \cos x}{\cos^4 x}\).

At \(x = a\), set \(\frac{dy}{dx} = 12\):

\(12 = \frac{\cos^2 a + 2a \sin a \cos a}{\cos^4 a}\).

Simplify to obtain:

\(12 \cos^3 a = \cos a + 2a \sin a\).

\(\cos^3 a = \frac{\cos a + 2a \sin a}{12}\).

\(a = \cos^{-1} \left( \sqrt[3]{\frac{\cos a + 2a \sin a}{12}} \right)\).

(b) Calculate \(\cos 0.9 \approx 0.622\) and \(\cos 1 \approx 0.540\).

\(\cos 0.9 > 0.553\) and \(\cos 1 < 0.570\), so \(0.9 < a < 1\).

(c) Use the iterative formula:

\(a_{n+1} = \cos^{-1} \left( \sqrt[3]{\frac{\cos a_n + 2a_n \sin a_n}{12}} \right)\).

Starting with \(a_0 = 0.9\), iterate to find:

\(a_1 \approx 0.95\), \(a_2 \approx 0.9743\), \(a_3 \approx 0.9694\), \(a_4 \approx 0.9704\).

Final answer: \(a \approx 0.97\).