Question

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

• A. $\frac{1}{2}\sec \frac{\theta }{2}$
• B. $\frac{1}{2}\cos \frac{\theta }{2}$
• C. $\sec \frac{\theta }{2}$
• D. $\cos \frac{\theta }{2}$

Answer 1

Answer: (A)

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