Question

General solution of equation $1+\cot \theta =cosec\theta$ is

• A. $2n\pi \pm \frac{\pi }{4}+\frac{\pi }{4}$
• B. $n\pi +(-1{)}^n\frac{\pi }{4}+\frac{\pi }{4}$
• C. $2n\pi +\frac{\pi }{3},n\in I$
• D. none of these

Answer 1

The correct option is A $2n\pi \pm \frac{\pi }{4}+\frac{\pi }{4}$

$1+\cot \theta =cosec\theta$
$1+\frac{\cos \theta }{\sin \theta }=\frac{1}{\sin \theta }$
$\Rightarrow \sin \theta +\cos \theta =1$
$\Rightarrow \frac{1}{\sqrt{2}}\sin \theta +\frac{1}{\sqrt{2}}\cos \theta =\frac{1}{\sqrt{2}}$
$\cos (\theta -\frac{\pi }{4})=\cos \frac{\pi }{4}$
$\theta -\frac{\pi }{4}=2n\pi \pm \frac{\pi }{4}$

$\theta =2n\pi \pm \frac{\pi }{4}+\frac{\pi }{4}$