(a) Expand \((\cos^2 \theta + \sin^2 \theta)^2\):

\((\cos^2 \theta + \sin^2 \theta)^2 = \cos^4 \theta + 2\cos^2 \theta \sin^2 \theta + \sin^4 \theta\)

Since \(\cos^2 \theta + \sin^2 \theta = 1\), we have:

\(\cos^4 \theta + \sin^4 \theta = 1 - 2\cos^2 \theta \sin^2 \theta\)

Using the double angle identity \(\sin 2\theta = 2\sin \theta \cos \theta\), we get:

\(\cos^4 \theta + \sin^4 \theta = 1 - \frac{1}{2} \sin^2 2\theta\)

(b) Solve \(\cos^4 \theta + \sin^4 \theta = \frac{5}{9}\):

Set \(1 - \frac{1}{2} \sin^2 2\theta = \frac{5}{9}\)

\(\frac{1}{2} \sin^2 2\theta = 1 - \frac{5}{9} = \frac{4}{9}\)

\(\sin^2 2\theta = \frac{8}{9}\)

\(\sin 2\theta = \pm \frac{2\sqrt{2}}{3}\)

Find \(2\theta\):

\(2\theta = \sin^{-1}\left(\frac{2\sqrt{2}}{3}\right)\) or \(2\theta = 180^\circ - \sin^{-1}\left(\frac{2\sqrt{2}}{3}\right)\)

Calculate \(\theta\):

\(\theta = 35.3^\circ, 54.7^\circ, 125.3^\circ, 144.7^\circ\)