Question

General solution of the equation $4cot 2 \theta +t a n^{2}\theta =cot^{2} ⁡ \theta $ is (where, $n\in Z$ )

• A. $n\pi \pm\frac{\pi }{4}$
• B. $n\pi \pm\frac{\pi }{3}$
• C. $2n\pi \pm\frac{\pi }{3}$
• D. $2n\pi \pm\frac{\pi }{6}$

Answer 1

Answer: (A)

The given equation can be written as $\frac{2\left(1-\tan ^{2} \theta\right)}{\tan \theta}+\tan ^{2} \theta=\frac{1}{\tan ^{2} \theta}$
$\Rightarrow \frac{2\left(1-\tan ^{2} \theta\right)+\tan ^{3} \theta}{\tan \theta}=\frac{1}{\tan ^{2} \theta}$
$\Rightarrow 2\left(1-\tan ^{2} \theta\right) \tan \theta+\tan ^{4} \theta-1=0$
$\Rightarrow \left(1-\tan ^{2} \theta\right)\left(2 \tan \theta-1-\tan ^{2} \theta\right)=0$
$\Rightarrow \left(1-\tan ^{2} \theta\right)(1-\tan \theta)^{2}=0$
$\Rightarrow (1+\tan \theta)(1-\tan \theta)^{3}=0$
$\Rightarrow \quad \tan \theta=\pm 1$
$\theta=n \pi \pm \frac{\pi}{4}$