Question

If $\cot \theta -pq\tan \theta =p-q;p,q\ne 0$, then the general value of $\theta$ is

• A. $m\pi +{\tan }^{-1}\left(\frac{1}{q}\right),m\in Z$
• B. $m\pi +{\tan }^{-1}\left(\frac{-1}{p}\right),m\in Z$
• C. $m\pi +{\tan }^{-1}\left(\frac{1}{pq}\right),m\in Z$
• D. $m\pi +{\tan }^{-1}\left(\frac{1}{p}\right),m\in Z$

Answer 1

The correct option is D $m\pi +{\tan }^{-1}\left(\frac{1}{p}\right),m\in Z$

$\cot \theta -pq\tan \theta =p-q$
$\Rightarrow 1-pq{\tan }^2\theta =(p-q)\tan \theta$
$\Rightarrow pq{\tan }^2\theta +(p-q)\tan \theta -1=0$
$\Rightarrow p\tan \theta (q\tan \theta +1)-1(q\tan \theta +1)=0$
$\Rightarrow (q\tan \theta +1)(p\tan \theta -1)=0$
$\Rightarrow \tan \theta =\frac{-1}{q},\frac{1}{p}$
$\Rightarrow \theta ={\tan }^{-1}\left(\frac{-1}{q}\right),{\tan }^{-1}\left(\frac{1}{p}\right)$

So, the general solution is
$m\pi +{\tan }^{-1}\left(\frac{-1}{q}\right)$
or $m\pi +{\tan }^{-1}\left(\frac{1}{p}\right)$