Can this equation always be true? $|\cos(\ln x)-2\sin(\ln x)| \le |\cos(\ln x)|-|2\sin(\ln x)|$

Question

$$|\cos(\ln x)-2\sin(\ln x)| \le |\cos(\ln x)|-|2\sin(\ln x)|$$

I'm confusing with $$|x-y|\ge||x|-|y||$$

because if I put minus in to $\sin(\ln x)$

It would be $$|\cos(\ln x)+2\sin(-\ln x)|\le|\cos(\ln x)|+|2\sin(-\ln x)|$$

Then it end up with $$|x+y|\le|x|+|y|$$

Answer 1

The inequality is not true.

Let $\tan y=-\frac 1 2$. Then, $\cos y=-2 \sin y$. If $x=e^{y}$ the RHS is $0$ but LHS is not.