(a) Where the left side is defined, \[\begin{aligned}\sin2\theta(\operatorname{cosec}\theta-\sec\theta)&=2\sin\theta\cos\theta\left(\frac1{\sin\theta}-\frac1{\cos\theta}\right)\\&=2\cos\theta-2\sin\theta\\&=2\sqrt2\left(\cos\theta\cos\frac\pi4-\sin\theta\sin\frac\pi4\right)\\&=\sqrt8\cos(\theta+\pi/4).\end{aligned}\]

(b) Since \(\theta+\pi/4\in(\pi/4,3\pi/4)\), there is one solution: \[\theta=\cos^{-1}(1/\sqrt8)-\frac\pi4\approx\boxed{0.424}\text{ radians}.\]

(c) Apply part (a) with \(\theta=x/2\) to write the integrand as \(2\cos(x/2)-2\sin(x/2)\). Therefore \[\boxed{4\sin(x/2)+4\cos(x/2)+C}.\]