Question

The value of $\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)\left[\cos \left(\frac{\pi }{2^2}\right)+i\sin \left(\frac{\pi }{2^2}\right)\right]$ $\left[\cos \left(\frac{\pi }{2^3}\right)+i\sin \left(\frac{\pi }{2^3}\right)\right].....\infty$ is:

• A. $-1$
• B. $1$
• C. $0$
• D. $\sqrt{2}$
• E. $-\sqrt{2}$

Answer 1

Answer: (A)

$\left(\cos \frac{\pi }{2}+i\sin \frac{\pi }{2}\right)\left(\cos \frac{\pi }{2^2}+i\sin \frac{\pi }{2^2}\right)....\infty$ $=\cos \left(\frac{\pi }{2}+\frac{\pi }{2^2}+....\infty \right)+i\sin \left(\frac{\pi }{2}+\frac{\pi }{2^2}+....\infty \right)$ $=\cos \left(\frac{\pi }{2}.\frac{1}{1-\frac{1}{2}}\right)+i\sin \left(\frac{\pi }{2}.\frac{1}{1-\frac{1}{2}}\right)$ $=\cos \pi +i\sin \pi$ $=-1$