Question

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If $z=\frac{1}{1-\cos \theta+i \sin \theta}$ and $\theta$ is acute then the modulus and amplitude of $z$ respectively are
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MathematicsComplex NumberAP EAMCETAP EAMCET 2018 (24 Apr Shift 2)

Options:

A $\frac{1}{2} \operatorname{cosec} \frac{\theta}{2},-\left(\frac{\pi}{2}+\frac{\theta}{2}\right)$
B $\frac{1}{2} \operatorname{cosec} \frac{\theta}{2},\left(\frac{\pi}{2}+\frac{\theta}{2}\right)$
C $\frac{1}{2} \sec \frac{\theta}{2},-\left(\frac{\pi}{2}+\frac{\theta}{2}\right)$
D $\frac{1}{2} \sec \frac{\theta}{2},\left(\frac{\pi}{2}+\frac{\theta}{2}\right)$