I found out after a few tries that the periods of both $$(\sin{\theta})^0, (\tan{\theta})^0~\text{are $\pi$}$$
Whereas i was told by my prof that period of $(\sin{\theta})^0 +(\tan{\theta})^0$ is $\frac{\pi}2$ as an exception to the LCM rule while finding the periods of two functions
Why is this anomaly happening? Logically speaking shouldn't the period of sum of two functions having periods $\pi$ be $\pi$ too?
In this context, $0^0$ is undefined, so $(\tan\theta)^0$ is undefined at all multiples of $\pi$. And $\tan\theta$ is undefined at odd multiples of $\pi/2.$ So this function is constant except at all multiples of $\pi/2$, therefore its period is $\pi/2$ not $\pi$.