Question

If $1+cot\theta =\cos ec\theta$, then the general value of $\theta$ is

• A. $n\pi +\frac{\pi }{2}$
• B. $2n\pi –\frac{\pi }{2}$
• C. $2n\pi +\frac{\pi }{2}$
• D. None of these

Answer 1

The correct option is C

$2n\pi +\frac{\pi }{2}$

Explanation for correct option :

Step 1: Apply trigonometric relation

$\Rightarrow 1+\cot \theta =cosec\theta$

$\Rightarrow 1+\frac{cos\theta }{\sin \theta }=\frac{1}{sin\theta }$ $\left[\because cot\theta =\frac{\cos \theta }{\sin \theta }\right]$

$\Rightarrow \frac{\sin \theta +\cos \theta }{\sin \theta }=\frac{1}{\sin \theta }$

$\Rightarrow \sin \theta +\cos \theta =1$

Step 2:Squaring both sides

$\Rightarrow {(\sin \theta +\cos \theta )}^2={\left(1\right)}^2$

$\Rightarrow {\sin }^2\theta +{\cos }^2\theta +2\sin \theta \cos \theta =1$ ; $\left[\because {\sin }^2\theta +{\cos }^2\theta =1\right]$

$\Rightarrow 1+2\sin \theta \cos \theta =1$ ; $\left[\because 2\sin \theta \cos \theta =\sin 2\theta \right]$

$\therefore \sin 2\theta =0$

$\Rightarrow \sin 2\theta =\sin 0=\sin \pi$

$\therefore \theta =2n\pi +\frac{\pi }{2}$

Hence Option (C) is correct.