(a) Start with the equation:

\(\sqrt{5} \sec x + \tan x = 4\)

Express \(\sec x\) and \(\tan x\) in terms of \(\cos x\) and \(\sin x\):

\(\sqrt{5} \frac{1}{\cos x} + \frac{\sin x}{\cos x} = 4\)

Multiply through by \(\cos x\):

\(\sqrt{5} + \sin x = 4 \cos x\)

Rearrange to:

\(4 \cos x - \sin x = \sqrt{5}\)

Use the identity:

\(a \cos \theta \pm b \sin \theta \equiv R \cos(\theta \mp \alpha)\)

where \(R = \sqrt{a^2 + b^2}\) and \(\tan \alpha = \frac{b}{a}\).

Here, \(a = 4\) and \(b = -1\):

\(R = \sqrt{4^2 + (-1)^2} = \sqrt{17}\)

\(\tan \alpha = \frac{-1}{4}\)

\(\alpha = \tan^{-1}\left(-\frac{1}{4}\right) \approx 14.04^\circ\)

(b) Solve \(\sqrt{5} \sec 2x + \tan 2x = 4\):

Using the result from (a):

\(4 \cos 2x - \sin 2x = \sqrt{5}\)

\(\cos(2x + 14.04^\circ) = \frac{\sqrt{5}}{\sqrt{17}}\)

\(2x + 14.04^\circ = \cos^{-1}\left(\frac{\sqrt{5}}{\sqrt{17}}\right)\)

Calculate \(\cos^{-1}\left(\frac{\sqrt{5}}{\sqrt{17}}\right) \approx 35.64^\circ\)

\(2x = 35.64^\circ - 14.04^\circ\)

\(2x = 21.6^\circ\)

\(x = 10.8^\circ\)

For the second solution:

\(2x = 180^\circ - 35.64^\circ + 14.04^\circ\)

\(2x = 158.4^\circ\)

\(x = 79.2^\circ\)

Thus, the solutions are \(x = 21.6^\circ\) and \(x = 144.4^\circ\).