Let $$z = \frac{{\cos \theta + i\sin \theta }}{{\cos \theta - i\sin \theta }},\frac{\pi }{4} < \theta < \frac{\pi }{2}.$$       Then $$\arg z$$  is

Solution :
$$z = {\left( {\cos \theta + i\sin \theta } \right)^2} = \cos 2\theta + i\sin 2\theta ,\,{\text{where }}\frac{\pi }{2} < 2\theta < \pi .$$
∴ $$z$$ is a complex number in the second quadrant. So, $$\frac{\pi }{2} < \arg \,z < \pi .$$
$$\therefore \,\,\arg\,z = {\tan ^{ - 1}}\left( {\tan 2\theta } \right) = 2\theta .$$