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

Question

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

Aicharya · Accepted Answer

Given, $	an 	heta=\cot 5 	heta$ $ \begin{aligned} & \Rightarrow \quad 	an 	heta=	an \left(\frac{\pi}{2}-5 	hetaight) \ & \Rightarrow \quad \frac{\pi}{2}-5 	heta=n \pi+	heta \ & \Rightarrow \quad 6 	heta=\frac{\pi}{2}-n \pi \ & \Rightarrow \quad 	heta=\frac{\pi}{12}-\frac{n \pi}{6} \ & 	ext { Also, } \cos 4 	heta=\sin 2 	heta=\cos \left(\frac{\pi}{2}-2 	hetaight) \ & \Rightarrow \quad 4 	heta=2 n \pi \pm\left(\frac{\pi}{2}-2 	hetaight) \end{aligned} $ Taking positive $ 6 	heta=2 n \pi+\frac{\pi}{2} \Rightarrow 	heta=\frac{n \pi}{3}+\frac{\pi}{12} $ Taking negative $ 2 	heta=2 n \pi-\frac{\pi}{2} \Rightarrow 	heta=n \pi-\frac{\pi}{4} $ Above values of $	heta$ suggests that there are only 3 common solutions

The number of values of $\theta$ in the interval $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ such that…

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