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What is the modulus of the complex number \(\rm \frac {\cos 2\theta - i \sin 2\theta}{\cos 2\theta + i \sin 2\theta}\) where \(\rm i = \sqrt {-1}\) ?
1. 1
2. 1/2
3. 3/2
4. None of these

1 Answer

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Best answer
Correct Answer - Option 1 : 1

Concept: 

Modulus of complex no. z =  a + ib is given by |z| = \(\rm \sqrt{a^{2}+ b^{2}}\) . 

Property of complex number:

\(\rm \left|\frac {z_1}{z_2}\right| = \frac {|z_1|}{|z_2|}\)

Calculation:

Let z = \(\rm \frac {\cos 2θ - i \sin 2θ}{\cos 2θ + i \sin 2θ}\)

Taking modulus on both sides, we get

⇒ |z| = \(\rm \left|\frac {\cos 2θ - i \sin 2θ}{\cos 2θ + i \sin 2θ}\right|\)

\(\rm \frac {|\cos 2θ - i \sin 2θ|}{|\cos 2θ + i \sin 2θ|}\)

\(\rm \frac {\sqrt {\cos^2 2\theta + \sin^2 2\theta}}{\sqrt {\cos^2 2\theta + \sin^2 2\theta}}\)

= 1

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