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 $2$

$\sin \theta +\cos \theta =p.............................\left(1\right)$

$\tan \theta +\cot \theta =q.............................\left(2\right)$

Squaring on both sides of $\left(1\right)$,

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

$\Rightarrow \sin 2\theta =p^2-1$

$\Rightarrow \csc 2\theta =\frac{1}{p^2-1}..........\left(3\right)$

From $\left(2\right)$,

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

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

$\csc 2\theta =\frac{q}{2}........\left(4\right)$

From $\left(3\right)$ and $\left(4\right)$,

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

Hence, $2$ is the right answer.