Trigonometry inequation with cotangent

Question

The second term equals $-\tan x$ and for the 3rd term I used the identity of $\cot(a+b)$. I obtain $\sqrt{3}\cot^3 x+3\cot^2 x-3\sqrt{3}\cot x-1$ How can I factorize this and solve the inequation?

Answer 1

Instead of using the formula for $\cot(a+b)$, let us use $$\cot a+\cot b=\dfrac{\sin(a+b)}{\sin a\sin b}$$ $\cot x+\cot(x+\frac{\pi}{2})=\dfrac{\cos 2x}{\sin x\cos x}=2\cot(2x)$

We obtain $2\cot (2x)+2\cot(x+\frac{\pi}{3})=2\left(\cot (2x)+\cot(x+\frac{\pi}{3})\right)$. Use the same identity on the last parenthesis.

Answer 2

Hint:

Use the identity $\cot3x=\dfrac{\cot^3x-3\cot x}{3\cot^2x-1}$.