What is the modulus of z?

Consider the following for the next three (03) items that follow : Let z=\frac{1+i~\sin~\theta}{1-i~\sin~\theta} where i=\sqrt{-1}

  1. A. 1 ✓
  2. B. \sqrt{2}
  3. C. 1+\sin^{2}\theta
  4. D. \frac{1+\sin^{2}\theta}{1-\sin^{2}\theta}

Correct Answer: A. 1

Explanation

The modulus of a complex number of the form \frac{a+ib}{c+id} is \frac{\sqrt{a^2+b^2}}{\sqrt{c^2+d^2}}. Here, |z| = \frac{|1+i\sin\theta|}{|1-i\sin\theta|} = \frac{\sqrt{1^2+\sin^2\theta}}{\sqrt{1^2+(-\sin\theta)^2}} = 1.

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