The number of values of theta in the interval | vedclass

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Question

$ 	an 	heta=\cot 5 	heta $

$ \Rightarrow \cos 6 	heta=0 $

$ 4 \cos ^3 2 	heta-3 \cos 2 	heta=0 $

$ \Rightarrow \cos 2 	heta=0 	ext {

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$ 	an 	heta=\cot 5 	heta $

$ \Rightarrow \cos 6 	heta=0 $

$ 4 \cos ^3 2 	heta-3 \cos 2 	heta=0 $

$ \Rightarrow \cos 2 	heta=0 	ext { or } \pm \frac{\sqrt{3}}{2} $

$ \sin 2 	heta=\cos 4 	heta $

$ \Rightarrow 2 \sin ^2 2 	heta+\sin 2 	heta-1=0 $

$ 2 \sin ^2 2 	heta+2 \sin 2 	heta-\sin 2 	heta-1=0 $

$ \sin 2 	heta=-1 	ext { or } \sin 2 	heta=\frac{1}{2} $

$ \cos 2 	heta=0 	ext { and } \sin 2 	heta=-1 $

$ \Rightarrow 2 	heta=-\frac{\pi}{2} \Rightarrow 	heta=-\frac{\pi}{4} $

$ \cos 2 	heta= \pm \frac{\sqrt{3}}{2}, \sin 2 	heta=\frac{1}{2} $

$ \Rightarrow 2 	heta=\frac{\pi}{6}, \frac{5 \pi}{6} \Rightarrow 	heta=\frac{\pi}{12}, \frac{5 \pi}{12} $

$ 	herefore 	heta=-\frac{\pi}{4}, \frac{\pi}{12}, \frac{5 \pi}{12}$

The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that $\theta \neq \frac{n \pi}{5}$ for $n=0, \pm 1, \pm 2$ and $\tan \theta=\cot 5 \theta$ as well as $\sin 2 \theta=\cos 4 \theta$ is

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