If {x_r} = cos (pi /{3^r}) - isin (pi / | vedclass

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Question

${{\rm{x}}_1} \cdot {{\rm{x}}_2} \cdot {{\rm{x}}_3}................\infty $

$ \Rightarrow \left( {\cos \frac{\pi }{3} - i\sin \frac{\pi }{3}} \righ

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$-i$

Vedclass · Answer

${{\rm{x}}_1} \cdot {{\rm{x}}_2} \cdot {{\rm{x}}_3}................\infty $

$ \Rightarrow \left( {\cos \frac{\pi }{3} - i\sin \frac{\pi }{3}} \right)\left( {\cos \frac{\pi }{9} - i\sin \frac{\pi }{9}} \right).........\infty $

$\Rightarrow \cos \left(\frac{\pi}{3}+\frac{\pi}{9}+\frac{\pi}{27}+\cdots\right)-i \sin \left(\frac{\pi}{3}+\frac{\pi}{9}+\frac{\pi}{27}+\cdots \infty\right)$

$\Rightarrow \cos \frac{\pi}{2}-i \sin \frac{\pi}{2}=-i$

If ${x_r} = \cos (\pi /{3^r}) - i\sin (\pi /{3^r}),$ (where $i = \sqrt{-1}),$ then value of $x_1.x_2.x_3......\infty ,$ is :-

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