{left( frac{cos theta + isin theta}{sin theta | vedclass

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(D) We have the expression $Z = {\left( \frac{\cos 	heta + i\sin 	heta}{\sin 	heta + i\cos 	heta} ight)^4}$.Note that $\sin 	heta + i\cos 	het

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$\cos 8	heta + i\sin 8	heta$

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(D) We have the expression $Z = {\left( \frac{\cos 	heta + i\sin 	heta}{\sin 	heta + i\cos 	heta} ight)^4}$.Note that $\sin 	heta + i\cos 	heta = i(\cos 	heta - i\sin 	heta)$.Thus,$Z = \frac{(\cos 	heta + i\sin 	heta)^4}{i^4(\cos 	heta - i\sin 	heta)^4}$.Since $i^4 = 1$,we use De Moivre's Theorem: $(\cos 	heta + i\sin 	heta)^n = \cos n	heta + i\sin n	heta$.$Z = \frac{\cos 4	heta + i\sin 4	heta}{\cos 4	heta - i\sin 4	heta}$.Multiply the numerator and denominator by the conjugate $(\cos 4	heta + i\sin 4	heta)$:$Z = \frac{(\cos 4	heta + i\sin 4	heta)^2}{\cos^2 4	heta + \sin^2 4	heta}$.Since $\cos^2 4	heta + \sin^2 4	heta = 1$,we get $Z = (\cos 4	heta + i\sin 4	heta)^2 = \cos 8	heta + i\sin 8	heta$.

${\left( \frac{\cos \theta + i\sin \theta}{\sin \theta + i\cos \theta} \right)^4}$ equals

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