The values of θ∈
Question:

The values of $\theta \in\left[0, \frac{\pi}{4}\right]$ satisfying $\tan 5 \theta=\cot 2 \theta$ are ___________________

Solution:

for $\theta \in\left[0, \frac{\pi}{4}\right]$

$\tan 5 \theta=\cot 2 \theta$

i.e $\tan 5 \theta=\frac{1}{\tan 2 \theta}$

$\Rightarrow \tan 5 \theta \tan 2 \theta=1 \quad \ldots(1)$

Since

$\tan (x+y)=\frac{\tan x+\tan y}{1-\tan x \tan y}$

i.e $\tan (5 \theta+2 \theta)=\frac{\tan 5 \theta+\tan 2 \theta}{1-\tan 5 \theta \tan 2 \theta}$

i.e $\tan 7 \theta=\frac{\tan 5 \theta+\tan 2 \theta}{0} \quad($ from 1$)$

i.e $\tan 7 \theta$ is not defined

i.e $\tan 7 \theta=\tan \frac{\pi}{2}$

$\Rightarrow 7 \theta=n \pi+\frac{\pi}{2}$

i.e $\theta=\frac{n \pi}{7}+\frac{\pi}{14}$

If $\theta \in\left[0, \frac{\pi}{4}\right], \theta=\frac{\pi}{14}, \frac{3 \pi}{14}$