Explanation

p=cosec(theta)-sin(theta)=cos^2(theta)/sin(theta), so p*sin(theta)=cos^2(theta). Similarly q=sec(theta)-cos(theta)=sin^2(theta)/cos(theta) gives q*cos(theta)=sin^2(theta). Sum: cos^2(theta)+sin^2(theta)=1. Writing p,q with common denominators is the move.