If frac{1+i cos theta}{1-2 i cos theta} is pu | vedclass

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(C) Let $Z = \frac{1+i \cos 	heta}{1-2 i \cos 	heta}$.To make $Z$ purely real,we multiply the numerator and denominator by the conjugate of the deno

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$2$

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(C) Let $Z = \frac{1+i \cos 	heta}{1-2 i \cos 	heta}$.To make $Z$ purely real,we multiply the numerator and denominator by the conjugate of the denominator,$(1+2i \cos 	heta)$:$Z = \frac{(1+i \cos 	heta)(1+2i \cos 	heta)}{(1-2i \cos 	heta)(1+2i \cos 	heta)}$$Z = \frac{1 + 2i \cos 	heta + i \cos 	heta + 2i^2 \cos^2 	heta}{1 + 4 \cos^2 	heta}$Since $i^2 = -1$,we have:$Z = \frac{(1 - 2 \cos^2 	heta) + i(3 \cos 	heta)}{1 + 4 \cos^2 	heta}$For $Z$ to be purely real,the imaginary part must be zero:$\operatorname{Im}(Z) = \frac{3 \cos 	heta}{1 + 4 \cos^2 	heta} = 0$This implies $\cos 	heta = 0$.If $\cos 	heta = 0$,then $\sin^2 	heta = 1 - \cos^2 	heta = 1$.Substituting these values into the expression:$\cos^3 	heta + \sin^2 	heta + \cos 	heta + 1 = (0)^3 + 1 + 0 + 1 = 2$.

If $\frac{1+i \cos \theta}{1-2 i \cos \theta}$ is purely real,then $\cos ^3 \theta+\sin ^2 \theta+\cos \theta+1=$

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