(a) Wherever the original expressions are defined, \[2\sin\theta\operatorname{cosec}2\theta=\frac{2\sin\theta}{2\sin\theta\cos\theta}=\frac1{\cos\theta}=\sec\theta.\]

(b) Put \(s=\sec\theta\). Then \(s^2-1+\frac72s=8\), so \[2s^2+7s-18=0\Rightarrow s=\frac{-7\pm\sqrt{193}}4.\] Both values have \(|s|\ge1\). Taking \(\theta=\pm\arccos(1/s)\) gives \(\boxed{\theta=\pm0.952,\ \pm1.76}\) radians, all in the stated interval and in the original domain.

(c) Using \(\sin x=2\sin(x/2)\cos(x/2)\), \[8\sin^2(x/2)\operatorname{cosec}^2x=2\sec^2(x/2).\] Hence the integral is \(4\tan(x/2)+C\), on intervals where the original integrand is defined.