(a) \(\cos^4\theta-\sin^4\theta=(\cos^2\theta-\sin^2\theta)(\cos^2\theta+\sin^2\theta)=\cos2\theta\). Also \(4\sin^2\theta\cos^2\theta=\sin^22\theta=1-\cos^22\theta\). Subtraction proves the identity.

(b) Write \(c=\cos2\alpha\). Then \(c^2+c-1=0\), so \(c=(-1\pm\sqrt5)/2\). Reject the root below −1. Thus \(2\alpha=51.8273^\circ,308.1727^\circ\), giving \(\alpha=25.9^\circ,154.1^\circ\).