$$\sin A + 2 \sin 2A + \sin 3A\,$$     is equal to which of the following ?
$$\eqalign{ & 1.\,\,\,\,4\sin 2A\,{\cos ^2}\left( {\frac{A}{2}} \right) \cr & 2.\,\,\,\,2\sin 2A{\left( {\sin \frac{A}{2} + \cos \frac{A}{2}} \right)^2} \cr & 3.\,\,\,\,8\sin A\cos A\,{\cos ^2}\left( {\frac{A}{2}} \right) \cr} $$
Select the correct answer using the code given below :

Solution :
$$\eqalign{ & {\text{Let}}\,A = {30^ \circ } \cr & \Rightarrow \sin A + 2\sin 2A + \sin 3A \cr & = \sin {30^ \circ } + 2\sin {60^ \circ } + \sin {90^ \circ } \cr & = \frac{1}{2} + \frac{{2\sqrt 3 }}{2} + 1 = \frac{{2\sqrt 3 + 3}}{2}\left( {\because 2\,{{\cos }^2}A = 1 + \cos 2A} \right) \cr & {\text{Now, }}4\sin 2A{\cos ^2}\left( {\frac{A}{2}} \right) = 2\sin 2A\left[ {1 + \cos A} \right] \cr & = 2\sin {60^ \circ }\left[ {1 + \cos {{30}^ \circ }} \right] = \frac{{2\sqrt 3 + 3}}{2} \cr & {\text{Also, }}\sin 2A = 2\sin A\cos A\,\,\& \,\,{\sin ^2}A + {\cos ^2}A = 1 \cr & 2\sin 2A{\left[ {\sin \frac{A}{2} + \cos \frac{A}{2}} \right]^2} \cr & = 2\sin 2A\left[ {{{\sin }^2}\frac{A}{2} + {{\cos }^2}\frac{A}{2} + 2\sin \frac{A}{2}\cos \frac{A}{2}} \right] \cr & = 2\sin 2A\left[ {1 + \sin A} \right] = 2\sin {60^ \circ }\left[ {1 + \sin {{30}^ \circ }} \right] = \frac{{3\sqrt 3 }}{2} \cr & \& \,\,8\sin A\cos A{\cos ^2}\left( {\frac{A}{2}} \right) \cr & = 4\sin A\cos A\left[ {1 + \cos A} \right] \cr & = 4\sin {30^ \circ }\cos {30^ \circ }\left[ {1 + \cos {{30}^ \circ }} \right] \cr & = \frac{{2\sqrt 3 + 3}}{2} \cr} $$