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

$\theta \in \left(-\dfrac{\pi}{2}, \dfrac{\pi}{2}\right)$ $\theta \neq \dfrac{n\pi}{5}, \quad n = 0, \pm 1, \pm 2$ $\tan\theta = \cot 5\theta$ $\sin 2\theta = \cos 4\theta$ $\text{To find: the number of such } \theta$ $\text{1. Solve } \tan\theta = \cot 5\theta$ $\text{2. Solve } \sin 2\theta = \cos 4\theta$ $\text{3. Take common values, drop } \theta = \dfrac{n\pi}{5}$ $\cot A = \tan\!\left(\dfrac{\pi}{2} - A\right)$ $\cot 5\theta = \tan\!\left(\dfrac{\pi}{2} - 5\theta\right)$ $\tan\theta = \tan\!\left(\dfrac{\pi}{2} - 5\theta\right)$ $\theta = \dfrac{\pi}{2} - 5\theta + k\pi$ $6\theta = \dfrac{\pi}{2} + k\pi$ $\theta = \dfrac{\pi}{12} + \dfrac{k\pi}{6}$ $k=0:\ \dfrac{\pi}{12};\quad k=1:\ \dfrac{\pi}{4};\quad k=2:\ \dfrac{5\pi}{12}$ $k=-1:\ -\dfrac{\pi}{12};\quad k=-2:\ -\dfrac{\pi}{4};\quad k=-3:\ -\dfrac{5\pi}{12}$ $\theta \in \left\{\pm\dfrac{\pi}{12},\ \pm\dfrac{\pi}{4},\ \pm\dfrac{5\pi}{12}\right\}$ $\Rightarrow 6 \text{ values from the first equation}$\cos 2x = 1 - 2\sin^2 x$$\cos 4\theta = 1 - 2\sin^2 2\theta$$\sin 2\theta = 1 - 2\sin^2 2\theta$$2\sin^2 2\theta + \sin 2\theta - 1 = 0$$2s^2 + s - 1 = 0,\quad s = \sin 2\theta$$2s^2 + s - 1 = (2s - 1)(s + 1) = 0$$\sin 2\theta = \dfrac{1}{2} \quad \text{or} \quad \sin 2\theta = -1$$2\theta = \dfrac{\pi}{6} + 2k\pi \ \text{ or } \ \dfrac{5\pi}{6} + 2k\pi$$\theta = \dfrac{\pi}{12} + k\pi \ \text{ or } \ \dfrac{5\pi}{12} + k\pi$$\theta = \dfrac{\pi}{12},\ \dfrac{5\pi}{12}$$2\theta = -\dfrac{\pi}{2} + 2k\pi$$\theta = -\dfrac{\pi}{4} + k\pi \ \Rightarrow\ \theta = -\dfrac{\pi}{4}$$\sin 2\theta = \cos 4\theta \Rightarrow \theta = -\dfrac{\pi}{4},\ \dfrac{\pi}{12},\ \dfrac{5\pi}{12}$$\text{Common: } \theta = -\dfrac{\pi}{4},\ \dfrac{\pi}{12},\ \dfrac{5\pi}{12}$$\text{Excluded: } \theta = 0,\ \pm\dfrac{\pi}{5},\ \pm\dfrac{2\pi}{5}$$-\dfrac{\pi}{4},\ \dfrac{\pi}{12},\ \dfrac{5\pi}{12} \ \notin\ \left{0,\ \pm\dfrac{\pi}{5},\ \pm\dfrac{2\pi}{5}\right}$$\text{Number of values of } \theta = 3$