Question

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

Answer 1

$cot\theta +tan\theta =2cosec\theta$

$\frac{cos\theta }{sin\theta }+\frac{sin\theta }{cos\theta }=\frac{2}{sin\theta }$

$\frac{co{s}^2\theta +si{n}^2\theta }{sin\theta cos\theta }=\frac{2}{sin\theta }$

$\frac{1}{sin\theta cos\theta }=\frac{2}{sin\theta }$

Now cross multiply,

$sin\theta =2sin\theta cos\theta$

$sin\theta -2sin\theta cos\theta =0$

$sin\theta (1-2cos\theta )=0$

$sin\theta =0$ or $cos\theta =\frac{1}{2}=cos\frac{\pi }{3}$

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

where $n$ is integer.

Therefore,

$\theta =\left(n\pi \right)\cup (2n\pi \pm \frac{\pi }{3})$