The number of distinct solutions of $$\sin 5\theta \cdot \cos 3\theta = \sin 9\theta \cdot \cos 7\theta $$      in $$\left[ {0,\frac{\pi }{2}} \right]$$  is

Solution :
$$\eqalign{ & \sin 8\theta + \sin 2\theta = \sin 16\theta + \sin 2\theta \,\,\,{\text{or, }}\sin 16\theta = \sin 8\theta \cr & \therefore \,\,16\theta = n\pi + {\left( { - 1} \right)^n}8\theta \cr & \Rightarrow \,\,8\theta = 2m\pi ,\,{\text{when }}n\,{\text{is even}} \cr & 24\theta = \left( {2m + 1} \right)\pi ,\,{\text{when }}n\,{\text{is odd}} \cr & \therefore \,\,\theta = \frac{{m\pi }}{4},\frac{{\left( {2m + 1} \right)\pi }}{{24}},\,{\text{when }}m \in {\Bbb Z} \cr & \theta = 0,\frac{\pi }{4},\frac{\pi }{2}\,{\text{and }}\frac{\pi }{{24}},\frac{\pi }{8},\frac{{5\pi }}{{24}},\frac{{7\pi }}{{24}},\frac{{3\pi }}{8},\frac{{11\pi }}{{24}}. \cr} $$