Question

Choose the correct answer from the alternatives given.
If $\sin \theta ,+\cos \theta =p$ and $\sec \theta ,+\text{cosec}\theta =q$. then $q(p^2$ $-1)$ = ?

• A. $p$
• B. $2p$
• C. $3p$
• D. $2p^2$

Answer 1

The correct option is B $2p$

$q(p^2-1)=(\sec \theta +\text{cosec}\theta )(\sin \theta +\cos \theta {)}^2-1)$

$=(\frac{1}{\cos \theta }+\frac{1}{\sin \theta })\{{\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta -1\}$

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

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

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