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

Concept:\(\sin {\RM{x}} + \sin {\rm{y}} = 2\sin \left( {\frac{{{\rm{x}} + {\rm{y}}}}{2}} \right)\cos \left( {\frac{{{\rm{x}} - {\rm{y}}}}{2}} \right)\)sin 2A = 2 sin A cos A\(1 + \cos {\rm{A}} = {\rm{\;}}2{\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right)\) Calculation:We have to find the value of Sin A + 2 sin 2 A + sin 3A⇒ sin A + 2 sin 2A + sin 3A = (sin A + sin 3A) + 2sin 2A= 2 sin 2A cos A + 2 sin 2A \(\left[\because {\sin {\rm{x}} + \sin {\rm{y}} = 2\sin \left( {\frac{{{\rm{x}} + {\rm{y}}}}{2}} \right)\cos \left( {\frac{{{\rm{x}} - {\rm{y}}}}{2}} \right)} \right]\)= 2 sin 2A (1 + cos A) ---(1)\(= 2{\rm{\;sin\;}}2{\rm{A}} \TIMES 2{\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right)\)\(= 4\sin 2{\rm{A}}{\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right)\)So, Statement (1) is correct.Now,\(= 4\sin 2{\rm{A}}{\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right) = 4 \times 2\sin {\rm{A}}\cos {\rm{A}} \times {\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right)\) (∵ sin 2A = 2 sin A cos A)\(= 8\sin {\rm{A}}\cos {\rm{A}}{\cos ^2}\left( {\frac{{\rm{A}}}{2}} \right)\)So, Statement (3) is also correct.Now, Check statement 2:\(2\sin 2{\rm{A\;}}{\left( {\sin \frac{{\rm{A}}}{2} + \cos \frac{{\rm{A}}}{2}} \right)^2} = 2\sin 2{\rm{A\;}}\left( {{{\sin }^2}\frac{{\rm{A}}}{2} + {{\cos }^2}\frac{{\rm{A}}}{2} + 2\sin \frac{{\rm{A}}}{2}\cos \frac{{\rm{A}}}{2}} \right) = 2\sin 2{\rm{A}}\left( {1 + \sin {\rm{A}}} \right)\)From EQUATION 1st,sin A + 2 sin 2A + sin 3A = 2 sin 2A (1 + cos A)So we can SAY that statement 2 is wrong.