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

Given, \(\tan \theta=\cot 5 \theta\)
\(\Rightarrow \tan \theta=\tan \left(\frac{\pi}{2}-5 \theta\right) \)
\( \Rightarrow \frac{\pi}{2}-5 \theta=n \pi+\theta \)
\( \Rightarrow 6 \theta=\frac{\pi}{2}-n \pi \)
\( \Rightarrow \theta=\frac{\pi}{12}-\frac{n \pi}{6} \)
\( \text {Also, } \cos 4 \theta=\sin 2 \theta=\cos \left(\frac{\pi}{2}-2 \theta\right) \)
\( \Rightarrow 4 \theta=2 n \pi \pm\left(\frac{\pi}{2}-2 \theta\right)\)
Taking positive
\(
6 \theta=2 n \pi+\frac{\pi}{2} \Rightarrow \theta=\frac{n \pi}{3}+\frac{\pi}{12}
\)
Taking negative
\(
2 \theta=2 n \pi-\frac{\pi}{2} \Rightarrow \theta=n \pi-\frac{\pi}{4}
\)
Above values of \(\theta\) suggests that there are only 3 common solutions.