ISL 1966 Problem 18

Solve the equation $\frac{1}{\sin x}+\frac{1}{\cos x}=\frac{1}{p},$ where $p$ is a real parameter. Investigate for which values of $p$ solutions exist and how many solutions exist.

(Of course, the last question ''how many solutions exist'' should be understood as ''how many solutions exists modulo $2\pi$ ''.)