Odd nature of sine function. (Taylor series) Although, this might be silly question. I am just wondering what happens to the odd nature of $\sin \theta$ when I expand it about some   $ \pi/4 $. There are terms with even powers appearing as well. Thanks for clarifying.

The even terms appear because $\sin \theta$ isn't odd about the point $\pi/4$. That is, $\sin(\theta-\pi/4)$ isn't odd. In fact, it's neither even nor odd. If you do an expansion about the point $\pi/2$, you will only see even terms.
