A) \[{{2}^{n}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\]
B) \[{{2}^{n}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\]
C) \[{{2}^{n+1}}\cdot {{\cos }^{n}}\frac{x}{2}\cdot \sin \frac{nx}{2}\]
D) \[{{2}^{n+1}}\cdot si{{n}^{n}}\frac{x}{2}\cdot \cos \frac{nx}{2}\]
Correct Answer: A
Solution :
| [a] \[\sum\limits_{r=0}^{n}{^{n}{{C}_{r}}\sin \,rx=\operatorname{Im}\left( \sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{e}^{irx}}} \right)}\] |
| \[=\operatorname{Im}\left( \sum\limits_{r=0}^{n}{^{n}{{C}_{r}}{{\left( {{e}^{ix}} \right)}^{r}}} \right)=\operatorname{Im}\left( {{\left( 1+{{e}^{ix}} \right)}^{n}} \right)\] |
| \[=\operatorname{Im}{{(1+\cos \,x+i\,\sin \,x)}^{n}}\] |
| \[=\operatorname{Im}\,{{(2\,co{{s}^{2}}\frac{x}{2}+2i\,sin\frac{x}{2}.cos\frac{x}{2})}^{n}}\] |
| \[=\operatorname{Im}\,{{\left( 2\cos \frac{x}{2}\left( \cos \frac{x}{2}+i\,\,\sin \frac{x}{2} \right) \right)}^{n}}\] |
| \[={{2}^{n}}.{{\cos }^{n}}\frac{x}{2}.\sin \frac{nx}{2}\] |
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