If sinθ + cosθ = $p$ and secθ + cosecθ = $q$, then $q(p^2-1)$ equals:

$q=1$/cosθ + 1/sinθ = (sinθ + cosθ)/(sinθ·cosθ) = $p$/(sinθ·cosθ). Squaring the first equation gives sinθ·cosθ = $(p^2-1)/2$, so $q=2p/(p^2-1)$ and therefore $q(p^2-1)=2p$.