Question

If $cot\theta +\tan \theta =2\cos ec\theta$, then the general value of $\theta$ is

• A. $n\pi \pm \frac{\pi }{3}$
• B. $n\pi \pm \frac{\pi }{6}$
• C. $2n\pi \pm \frac{\pi }{3}$
• D. $2n\pi \pm \frac{\pi }{6}$

Answer 1

The correct option is C

$2n\pi \pm \frac{\pi }{3}$

Find the value of $\theta :$

Given, $cot\theta +\tan \theta =2\cos ec\theta$

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

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

$\Rightarrow$ $2\cos ec\theta =\frac{1}{\sin \theta \cos \theta }$ $\left[\mathbf{\therefore }\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\mathit{A}\mathbf{}\mathbf{+}\mathbf{}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\mathit{A}\mathbf{}\mathbf{=}\mathbf{}\mathbf{1}\right]$

$\Rightarrow$$2\cos ec\theta \sin \theta \cos \theta =1$

$\Rightarrow$ $2\cos \theta =1$ $\left[\mathbf{\because }\mathit{c}\mathit{o}\mathit{s}\mathit{e}\mathit{c}\mathbf{}\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{1}}{\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{}\mathbf{\theta }}\right]$

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

$\Rightarrow$ $\cos \theta =\cos \frac{\pi }{3}$

$\mathbf{\therefore }\mathit{\theta }\mathbf{}\mathbf{=}\mathbf{}\mathbf{2}\mathit{n}\mathit{\pi }\mathbf{}\mathbf{\pm }\mathbf{}\frac{\mathbf{\pi }}{\mathbf{3}}$

Hence, Option ‘C’ is Correct.