Help with trigonometric equation $\tan(x)+\cot(2x)=1$

Question

I am trying to solve the equation $\tan(x)+\cot(2x)=1$.

It is clear that the equation has a solution $x=\pi/4$ but I can't show that this is the only solution. Any ideas?

P.S. I am trying to solve the equation without using formulas for $2a$ angle

Answer 1

$\tan x=t$

$$1=t+\dfrac{1-t^2}{2t}$$

$$\iff2t=1+t^2\iff(t-1)^2=0$$

Answer 2

Actually, you don't have to use double angle formulas at all...

$$\tan (x) + \cot 2x = 1 \Rightarrow \dfrac {\sin x}{\cos x} + \dfrac {\cos 2x}{\sin 2x} =1 \Rightarrow \dfrac {\sin x \sin 2x + \cos x \cos 2x}{\cos x \sin 2x} = 1$$

A couple of hints:

1. What trick can you use to simplify ${\sin x \sin 2x + \cos x \cos 2x}$?
2. Are there any extraneous roots? ($\frac {\pi}{4}$ is correct; the general solution would be $\frac {\pi}{4} \pm \pi k$.)