Mathematics - Trigonometric Equation Question with Solution | TestHub

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