Why is the period of $(\sin{\theta})^0 +(\tan{\theta})^0$, $\frac{\pi}{2}$
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 $\pi$.