Question

solve the equation $3cosec{}^2\theta =2\sec \theta$

Answer 1

$3{\csc }^2\theta =2\sec \theta$

$\Rightarrow \frac{3}{{\sin }^2\theta }=\frac{2}{\cos \theta }$

$\Rightarrow 3\cos \theta =2{\sin }^2\theta$

$\Rightarrow 3\cos \theta =2(1-{\cos }^2\theta )$

$\Rightarrow 3\cos \theta =2-2{\cos }^2\theta$

$\Rightarrow 2{\cos }^2\theta +3\cos \theta -2=0$

$\Rightarrow 2{\cos }^2\theta +4\cos \theta -\cos \theta -2=0$

$\Rightarrow 2\cos \theta (\cos \theta +2)-1(\cos \theta +2)=0$

$\Rightarrow (\cos \theta +2)(2\cos \theta -1)=0$

$\Rightarrow \cos \theta =-2$ is not possible

$\therefore 2\cos \theta =1$

$\Rightarrow \cos \theta =\frac{1}{2}$ either in first or fourth quadrant.

$\Rightarrow \theta =2n\pi \pm \frac{\pi }{3}$