Answer: \(\theta=-101.9^\circ\) or \(101.9^\circ\).

Answer: \(\theta=-101.9^\circ\) or \(101.9^\circ\).

(a) Combine the two terms over a common denominator:

The numerator is

\(\cos^{2}x+1-2\sin x+\sin^{2}x=2-2\sin x=2(1-\sin x).\)

Therefore

(b) Let \(u=\frac12\theta\). By part (a), the equation becomes

\(2\operatorname{sec}u=8\cos^{2}u.\)

Since \(\cos u\ne0\), multiply by \(\cos u\):

\(2=8\cos^{3}u.\)

Hence

\(\cos^{3}u=\frac14,\qquad \cos u=\sqrt[3]{\frac14}.\)

Now \(-360^\circ\lt \theta\lt 360^\circ\), so \(-180^\circ\lt u\lt 180^\circ\). Thus

\(u=\pm 50.95^\circ\quad ext{to 2 decimal places}.\)

Therefore

\(\theta=2u=\pm101.9^\circ.\)