Question

If $\cot \theta +\tan \theta =2cosec\theta$, then find the general value of $\theta$.

Answer 1

Given:
$\cot \theta +\tan \theta =2cosec\theta$
$\Rightarrow \frac{\cos \theta }{\sin \theta }+\frac{\sin \theta }{\cos \theta }=\frac{2}{\sin \theta }$
$\left[{}_{cosec\theta =\frac{1}{\sin \theta }}^{\cot \theta =\frac{\cos \theta }{\sin \theta },\tan \theta =\frac{\sin \theta }{\cos \theta }\text{and}}\right]$
$\Rightarrow \frac{{\sin }^2\theta +{\cos }^2\theta }{\sin \theta \cos \theta }=\frac{2}{\sin \theta }$
$\Rightarrow \frac{1}{\cos \theta }=2(\sin \theta \ne 0$ and ${\sin }^2\theta +{\cos }^2\theta =1)$
$\Rightarrow \cos \theta =\frac{1}{2}$
$\Rightarrow \cos \theta =\cos \frac{\pi }{3}$
$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3},nϵZ$
$(\because \cos \theta =\cos \alpha \Rightarrow \theta =2n\pi \pm \alpha ,nϵZ)$
Hence, the general solution is
$\theta =2n\pi \pm \frac{\pi }{3},nϵZ$