For n∈Z+ 1+sin θ+i cos θn+1+sin θ-i cosθn==

(1+sinθ+i cosθ)n+(1+sinθ-i cosθ)n = 1+cosπ2-θ+i sinπ2-θn+1+cosπ2-θ-i sinπ2-θn2cos2π4-θ2+i2sinπ4-θ2cosπ4-θ2n+2cos2π4-θ2-i2sinπ4-θ2cosπ4-θ2n2cosπ4-θ2ncosπ4-θ2+isinπ4-θ2n+cosπ4-θ2-isinπ4-θ2n2cosπ4-θ2ncos nπ4-θ2+isin n π4-θ2+cos nπ4-θ2-isinn π4-θ22ncosπ4-θ2n2cosnπ4-θ2=2n+1.cosnπ4-θ2 cosnπ4-nθ2

For n∈Z+ 1+sin θ+i cos θn+1+sin θ-i cosθn==

(1+sinθ+i cosθ)n+(1+sinθ-i cosθ)n = 1+cosπ2-θ+i sinπ2-θn+1+cosπ2-θ-i sinπ2-θn2cos2π4-θ2+i2sinπ4-θ2cosπ4-θ2n+2cos2π4-θ2-i2sinπ4-θ2cosπ4-θ2n2cosπ4-θ2ncosπ4-θ2+isinπ4-θ2n+cosπ4-θ2-isinπ4-θ2n2cosπ4-θ2ncos nπ4-θ2+isin n π4-θ2+cos nπ4-θ2-isinn π4-θ22ncosπ4-θ2n2cosnπ4-θ2=2n+1.cosnπ4-θ2 cosnπ4-nθ2

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$For n \in Z^{+} {(1 + \sin θ + i \cos θ)}^{n} + {(1 + \sin θ - i cosθ)}^{n} =$ =

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$2^{n + 1} . \cos^{n} (\frac{π}{4} - \frac{θ}{2}) \cos (\frac{nπ}{4} - \frac{θ}{2})$

$2^{n + 1} . \cos^{n} (\frac{π}{4} - \frac{θ}{2}) \sin (\frac{nπ}{4} - \frac{θ}{2})$

$2^{n + 1} . \cos^{n} (\frac{π}{4} - \frac{θ}{2}) \cos (\frac{nπ}{4} - \frac{nθ}{2})$

$2^{n + 1} . \cos^{n} (\frac{π}{4} - \frac{θ}{2}) \sin (\frac{nπ}{4} - \frac{nθ}{2})$

answer is C.

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Detailed Solution

$\begin{array}{rcl} {(1 + sinθ + i cosθ)}^{n} + {(1 + sinθ - i cosθ)}^{n} =^{} \end{array}$

$\begin{array}{rcl} {(1 + \cos (\frac{π}{2} - θ) + i \sin (\frac{π}{2} - θ))}^{n} + {(1 + \cos (\frac{π}{2} - θ) - i \sin (\frac{π}{2} - θ))}^{n} \end{array}$

${(2 \cos^{2} (\frac{π}{4} - \frac{θ}{2}) + i 2 \sin (\frac{π}{4} - \frac{θ}{2}) \cos (\frac{π}{4} - \frac{θ}{2}))}^{n} + {(2 \cos^{2} (\frac{π}{4} - \frac{θ}{2}) - i 2 \sin (\frac{π}{4} - \frac{θ}{2}) \cos (\frac{π}{4} - \frac{θ}{2}))}^{n}$

${(2 \cos (\frac{π}{4} - \frac{θ}{2}))}^{n} {(\cos (\frac{π}{4} - \frac{θ}{2}) + i \sin (\frac{π}{4} - \frac{θ}{2}))}^{n} + {(c os (\frac{π}{4} - \frac{θ}{2}) - i \sin (\frac{π}{4} - \frac{θ}{2}))}^{n}$

${(2 \cos (\frac{π}{4} - \frac{θ}{2}))}^{n} (\cos n (\frac{π}{4} - \frac{θ}{2}) + i \sin n (\frac{π}{4} - \frac{θ}{2})) + (c os n (\frac{π}{4} - \frac{θ}{2}) - i \sin n (\frac{π}{4} - \frac{θ}{2}))$

$2^{n} {(\cos (\frac{π}{4} - \frac{θ}{2}))}^{n} (2 \cos n (\frac{π}{4} - \frac{θ}{2}))$ = $2^{n + 1} . \cos^{n} (\frac{π}{4} - \frac{θ}{2}) \cos (\frac{nπ}{4} - \frac{nθ}{2})$