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  4. If θ is real then the modulus of (1/1+ cos θ+i sin θ) is

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Q. If $\theta$ is real then the modulus of $\frac{1}{1+\cos \theta+i \sin \theta}$ is

Solution:

$\left|\frac{1}{1+\cos \theta+i \sin \theta}\right|=\frac{1}{2 \cos \frac{\theta}{2}}\left|\frac{1}{\cos \frac{\theta}{2}+i \sin \frac{\theta}{2}}\right|=\frac{1}{2 \cos \frac{\theta}{2}}=\frac{1}{2} \sec \frac{\theta}{2}$