I don't know why anyone would make you use x0=0 unless they wanted you to lose your hair. Think about where you are on the curve of cos(x). Think about where your tangent line is. Algebraically, you know that 1/0 is undefined, but geometrically, you're looking for the intersection of two parallel lines -- something you're not going to find either.
You should begin to see that the periodic nature of the sinusoids means that you'll need to pick a starting point close to a desired root, or else you'll end up overshooting and finding one way off down the line. Try x0=0.1 and then try x0=1.5.
Also, this is probably supposed to be x0 instead of p0.
fprintf('\nApproximate solution xn= %11.8f \n\n',p0);
