Question

If $\sin \theta +\cos \theta =p$ and $\sec \theta +cosec\theta =q$, show that $q(p^2-1)=2p$.

Answer 1

LHS :

$q(p^2-1)$

$=(\sec \theta +cosec\theta )\left[\right(\sin \theta +\cos \theta {)}^2-1]$

$=(\frac{1}{\cos \theta }+\frac{1}{\sin \theta })\left(2\sin \theta \cos \theta \right)$

$=\left(\frac{\sin \theta +\cos \theta }{\sin \theta \cos \theta }\right)\left(2\sin \theta \cos \theta \right)$

$=2(\sin \theta +\cos \theta )=2p$ = RHS