Does the function $ f (t)= \sin t + \log(\tan (\frac{t}{2})) $ have a derivative of all orders?

Question

Does the function $ f (t)= \sin t + \log(\tan (\frac{t}{2})) $ have a derivative of all orders?

I know that the composition of differentiable functions is differentiable and also the sine function is infinitely differentiable, but I am not sure about the function as a whole.

Answer 1

All the elementary functions in $f$ are infinitely differentiable over their domains, and they are composed in ways preserving that infinite differentiability. Hence $f$ must be infinitely differentiable over its domain, which turns out to be $\bigcup_{n\in\mathbb Z}(2\pi n,2\pi n+\pi)$.