How to manually find the solutions to this equation?

Question

In an optimization problem I finally get to the point where I have to solve

$$x +\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x)) =0$$

which obviously leads to

$$x=-\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x))$$

Nevertheless, this couldn't in any case help knowing the optimal size of the angle $x$. Wolfram Alpha can give me the solutions, but is not able to give me a step-by-step explanation of how to reach an approximate value for $x$.

Then, how do we solve manually such an equation?

Answer 1

We have that $\cos 2x=\cos^2x-\sin^2x=2\cos^2x-1,\sin 2x=2\sin x\cos x.$ Now

$$\tan(x)\cos(2x)+\tan(x)= \tan x(\cos 2x+1)=2\tan x\cos^2x=2\sin x\cos x.$$

Thus

$$\tan(x)\cos(2x)+\tan(x)-2\sin(2x)=-2\sin x\cos x.$$

So

$$\sec(x)(\tan(x)\cos(2x)+\tan(x)-2\sin(2x))=-2\sin x.$$

So, you equation is just $x=2\sin x.$

It is easy to see that $x=0$ is a solution. The other solutions are not so easy to get manually.

Answer 2

Hint: Use $\cos(2x) = 2\cos^2(x) - 1$ and $\sin(2x) = 2\sin(x) \cos(x)$

by the way do you get $x = 0$ and 2 other solutions (symmetric)?