If $z=\frac{1}{1-\cos \theta+i \sin \theta}$ and $\theta$ is acute then the modulus and amplitude of $z$ respectively are $$

• Correct answer
  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

  $\frac{1}{2} \sec \frac{\theta}{2},\left(\frac{\pi}{2}+\frac{\theta}{2}\right)$

Hence, the correct answer is option: 'A'.