Question

If $\sin \theta +\cos \theta =p$ and $\sec \theta +cosec\theta =q,0\le \theta \le \frac{\pi }{2},$ and $q(p^2-1)=X$, then

• A. $X>2$
• B. $X\ge 2$
• C. $X>4$
• D. $X\ge 4$

Answer 1

The correct option is B $X\ge 2$

$q(p^2-1)=(sec\theta +cosec\theta )\left(\right(sin\theta +cos\theta {)}^2-1)$
$=(sec\theta +cosec\theta )(si{n}^2\theta +co{s}^2\theta +2sin\theta cos\theta -1)$
$=(sec\theta +cosec\theta )(1+2sin\theta cos\theta -1)$ (Since, ${\sin }^2\theta +{\cos }^2\theta =1$)
$=(\frac{1}{cos\theta }+\frac{1}{sin\theta })\left(2sin\theta cos\theta \right)$
$=2sin\theta +2cos\theta$