Question

If $\cot \theta +\cos \theta =p$ and $\cot \theta -\cos \theta =q$, then the value of $p^2-q^2$ is

• A. $2\sqrt{pq}$
• B. $4\sqrt{pq}$
• C. $2pq$
• D. $4pq$

Answer 1

The correct option is B $4\sqrt{pq}$

$p^2-q^2$
$=(cot\theta +cos\theta {)}^2-(cot\theta -cos\theta {)}^2$
$=(co{s}^2\theta +co{t}^2\theta +2cos\theta cot\theta )-(co{s}^2+co{t}^2\theta -2cos\theta cot\theta )$
$=4cos\theta .cot\theta$
$=\frac{4co{s}^2\theta }{sin\theta }$
$=4\sqrt{\frac{co{s}^4\theta }{si{n}^2\theta }}$

$=4\sqrt{\frac{co{s}^2\theta \times co{s}^2\theta }{si{n}^2\theta }}$

$=4\sqrt{\frac{co{s}^2\theta \times (1-si{n}^2\theta )}{si{n}^2\theta }}$

$=4\sqrt{co{t}^2\theta -co{s}^2\theta }$
$=4\sqrt{(cot\theta +cos\theta )(cot\theta -cos\theta )}$
$=4\sqrt{pq}$