$$\sum\limits_{k = 33}^{65} {\left( {\sin \frac{{2k\pi }}{8} - i\cos \frac{{2k\pi }}{8}} \right)} $$

Solution :
$$\eqalign{ & \sum\limits_{k = 33}^{65} {\left( {\sin \frac{{2k\pi }}{8} - i\cos \frac{{2k\pi }}{8}} \right)} \cr & = \left[ {\sin \frac{{33\pi }}{4} + \sin \frac{{34\pi }}{4} + ..... + \sin \frac{{65\pi }}{4}} \right] - i\left[ {\cos \frac{{33\pi }}{4} + \cos \frac{{34\pi }}{4} + .... + \cos \frac{{65\pi }}{4}} \right] \cr & = \sin \frac{\pi }{4} - i\cos \frac{\pi }{4}\sin \alpha + \sin \left( {\alpha + \beta } \right) + \sin \left( {\alpha + 2\beta } \right) + ..... + \sin \left[ {\alpha + \left( {n - 1} \right)\beta } \right] \cr & = \frac{{\sin \left\{ {\alpha + \left( {n - 1} \right)\frac{\beta }{2}} \right\} \cdot \sin \frac{{n\beta }}{2}}}{{\sin \frac{\beta }{2}}} \cr & {\text{and }}\cos \left( \alpha \right) + \cos \left( {\alpha + \beta } \right) + ..... + \cos \left( {\alpha + \left( {n - 1} \right)\beta } \right) \cr & = \frac{{\cos \left\{ {\alpha + \left( {n - 1} \right)\frac{\beta }{2}} \right\} \cdot \sin \left( {\frac{{n\beta }}{2}} \right)}}{{\sin \frac{\beta }{2}}} \cr & = - \left( {\frac{{1 + i}}{{\sqrt 2 }}} \right) = \frac{{1 - i}}{{\sqrt 2 }} \cr} $$