Question

If $\sin$ $\theta +$ $\cos$ $\theta =p$ and $\tan$ $\theta +$ $\cot$ $\theta =q$ then $q(p^2-1)=$

• A. $\frac{1}{2}$
• B. $2$
• C. $1$
• D. $3$

Answer 1

The correct option is B $2$

($\sin \theta$ + $\cos \theta {)}^2=p^2$

$\Rightarrow 1+$ $\sin 2\theta =p^2$ [$\because$ sin $2\theta =p^2-1$]

$\tan \theta +\frac{1}{\tan \theta }=q$

$\Rightarrow \frac{{\tan }^2\theta +1}{2\tan \theta }=\frac{q}{2}$

$\text{cosec}2\theta =\frac{q}{2}$

$\Rightarrow \frac{1}{p^2-1}=\frac{q}{2}\Rightarrow 2=(p^2-1)q$