Question
The value of $$\sum\limits_{r = 0}^n {^n{C_r}\sin \left( {rx} \right)} $$ is equal to
A.
$${2^n} \cdot {\cos ^n}\frac{x}{2} \cdot \sin \frac{{nx}}{2}$$
B.
$${2^n} \cdot {\sin ^n}\frac{x}{2} \cdot \cos \frac{{nx}}{2}$$
C.
$${2^{n + 1}} \cdot {\cos ^n}\frac{x}{2} \cdot \sin \frac{{nx}}{2}$$
D.
$${2^{n + 1}} \cdot {\sin ^n}\frac{x}{2} \cdot \cos \frac{{nx}}{2}$$
Answer :
$${2^n} \cdot {\cos ^n}\frac{x}{2} \cdot \sin \frac{{nx}}{2}$$
Solution :
$$\eqalign{
& \sum\limits_{r = 0}^n {^n{C_r}\sin \left( {rx} \right)} = \operatorname{Im} \left( {\sum\limits_{r = 0}^n {^n{C_r}{e^{irx}}} } \right) \cr
& = \operatorname{Im} \left( {\sum\limits_{r = 0}^n {^n{C_r}{{\left( {{e^{ix}}} \right)}^r}} } \right) = \operatorname{Im} \left( {{{\left( {1 + {e^{ix}}} \right)}^n}} \right) \cr
& = \operatorname{Im} {\left( {1 + \cos x + i\sin x} \right)^n} \cr
& = \operatorname{Im} {\left( {2{{\cos }^2}\frac{x}{2} + 2i\sin \frac{x}{2} \cdot \cos \frac{x}{2}} \right)^n} \cr
& = \operatorname{Im} {\left( {2\cos \frac{x}{2}\left( {\cos \frac{x}{2} + i\sin \frac{x}{2}} \right)} \right)^n} \cr
& = {2^n} \cdot {\cos ^n}\frac{x}{2} \cdot \sin \frac{{nx}}{2} \cr} $$