Question

Solution of the equation $4\cot 2\theta ={\cot }^2\theta -{\tan }^2\theta$ is

• A. $\theta =n\pi \pm \frac{\pi }{2}$
• B. $\theta =n\pi \pm \frac{\pi }{3}$
• C. $\theta =n\pi \pm \frac{\pi }{4}$
• D. None of these

Answer 1

The correct option is C $\theta =n\pi \pm \frac{\pi }{4}$

We have, $4\cot 2\theta ={\cot }^2\theta -{\tan }^2\theta \Rightarrow \frac{4}{\tan 2\theta }=\frac{1}{{\tan }^2\theta }-{\tan }^2\theta$

Substitute $\tan 2\theta =\frac{2\tan \theta }{1-{\tan }^2\theta }$

$\therefore \frac{4(1-{\tan }^2\theta )}{2\tan \theta }=\frac{1-{\tan }^4\theta }{{\tan }^2\theta }$

$\Rightarrow (1-{\tan }^2\theta )[2\tan \theta -(1+{\tan }^2\theta )]=0$

$\Rightarrow (1-{\tan }^2\theta )({\tan }^2\theta -2\tan \theta +1)=0$

$\Rightarrow (1-{\tan }^2\theta ){(\tan \theta -1)}^2=0$

$\Rightarrow {(1-\tan \theta )}^3(1+\tan \theta )=0$

$\Rightarrow \tan \theta =1,-1$

$\Rightarrow \theta =n\pi \pm \frac{\pi }{4}$