The value of the expression left( cos frac{pi | vedclass

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(A) Using the property of complex numbers in polar form,$\left( \cos 	heta_1 + i\sin 	heta_1 ight) \left( \cos 	heta_2 + i\sin 	heta_2 ight) =

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$-1$

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(A) Using the property of complex numbers in polar form,$\left( \cos 	heta_1 + i\sin 	heta_1 ight) \left( \cos 	heta_2 + i\sin 	heta_2 ight) = \cos(	heta_1 + 	heta_2) + i\sin(	heta_1 + 	heta_2)$.Given expression $= \cos \left( \frac{\pi }{2} + \frac{\pi }{{{2^2}}} + \frac{\pi }{{{2^3}}} + \dots ight) + i\sin \left( \frac{\pi }{2} + \frac{\pi }{{{2^2}}} + \frac{\pi }{{{2^3}}} + \dots ight)$.The sum of the infinite geometric series in the argument is $S = \frac{\pi}{2} + \frac{\pi}{4} + \frac{\pi}{8} + \dots = \frac{\pi/2}{1 - 1/2} = \frac{\pi/2}{1/2} = \pi$.Thus,the expression becomes $\cos(\pi) + i\sin(\pi)$.Since $\cos(\pi) = -1$ and $\sin(\pi) = 0$,the value is $-1 + i(0) = -1$.

The value of the expression $\left( \cos \frac{\pi }{2} + i\sin \frac{\pi }{2} \right) \left( \cos \frac{\pi }{{{2^2}}} + i\sin \frac{\pi }{{{2^2}}} \right) \dots$ to $\infty$ is

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