The value of the infinite product (cos theta | vedclass

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(B) Using the property of complex numbers in polar form,$(\cos \alpha + i\sin \alpha)(\cos \beta + i\sin \beta) = \cos(\alpha + \beta) + i\sin(\alpha

Vedclass · Accepted Answer

$\cos 2	heta + i\sin 2	heta $

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(B) Using the property of complex numbers in polar form,$(\cos \alpha + i\sin \alpha)(\cos \beta + i\sin \beta) = \cos(\alpha + \beta) + i\sin(\alpha + \beta)$.Given the product is $P = (\cos 	heta + i\sin 	heta )(\cos \frac{	heta }{2} + i\sin \frac{	heta }{2})(\cos \frac{	heta }{2^2} + i\sin \frac{	heta }{2^2}) \dots \infty$.This simplifies to $P = \cos(	heta + \frac{	heta }{2} + \frac{	heta }{2^2} + \dots) + i\sin(	heta + \frac{	heta }{2} + \frac{	heta }{2^2} + \dots)$.The exponent sum is a geometric series with first term $a = 	heta$ and common ratio $r = 1/2$.The sum of the infinite series is $S = \frac{a}{1-r} = \frac{	heta}{1 - 1/2} = \frac{	heta}{1/2} = 2	heta$.Therefore,$P = \cos(2	heta) + i\sin(2	heta)$.

The value of the infinite product $(\cos \theta + i\sin \theta )(\cos \frac{\theta }{2} + i\sin \frac{\theta }{2})(\cos \frac{\theta }{2^2} + i\sin \frac{\theta }{2^2}) \dots$ is

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