Answer
See proof below.
Work Step by Step
(a) Given $f(t+p)=f(t)$, let $g(t)=\frac{1}{f(t)}$, we have $g(t+p)=\frac{1}{f(t+p)}=\frac{1}{f(t)}=g(t)$,
thus $g(t)$ is also periodic with period $p$
(b) Since $csc(t)=\frac{1}{sin(t)}$ and $sec(t)=\frac{1}{cos(t)}$ and we know that $sin(t), cos(t)$ both are periodic with period $2\pi$, use the results from (a), we know that $csc(t)$ and $sec(t)$ are also periodic with period $2\pi$.
