What is the modulus of z?

Concept:

If z = x + iy 

⇒ Modulus of z = 

Calculation:

Given, 

z =  where i = 

Multiply numerator and denominator by (1 + i sin θ),

,

\(⇒ |z| =\sqrt{(\frac{ \cos^2\theta }{1+\sin^2 θ})^2 +(\frac{2\sin θ}{1+\sin^2 θ})^2}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{\cos^4 \theta + 4\sin^2 \theta}\)

\(⇒ |z| = {1 \over 1+ \sin^2 \theta }\sqrt{(1 -\sin^2 \theta)^2 + 4\sin^2 \theta}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{1 -2\sin^2 \theta + \sin^4 \theta + 4\sin^2 \theta}\)

\(⇒ |z| ={1 \over 1+ \sin^2 \theta }\sqrt{1 +2\sin^2 \theta + \sin^4 \theta }\)

\(⇒ |z| ={1+ \sin^2 \theta \over 1+ \sin^2 \theta } = 1\)

∴ The correct answer is option (1).