(a) Wherever defined, \[\frac1{1-\sin\theta}+\frac1{1+\sin\theta}=\frac{(1+\sin\theta)+(1-\sin\theta)}{(1-\sin\theta)(1+\sin\theta)}=\frac2{1-\sin^2\theta}=\frac2{\cos^2\theta}.\]

(b) By (a), \(6/\cos^2\theta=12\), so \(\cos^2\theta=1/2\) and \(\cos\theta=\pm1/\sqrt2\). Thus \[\boxed{\theta=45^\circ,135^\circ,225^\circ,315^\circ}.\] All four satisfy the original denominators.