Adding $k\pi$ vs adding $2k\pi$ in solutions to trig inequalities involving tangent or cotangent

Question

An author usually adds $k\pi$ to solutions of a trigonometric inequality that includes tangent or cotangent. However, in this example, $$\log_{\tan{x}}{\sin{x}}>1$$ he used $2k\pi$, so that the solutions are: $$x\in\left(2k\pi,\frac{\pi}{4}+2k\pi\right)$$

What made him use $2k\pi$ instead of just $k\pi$?

Answer 1

It's most likely to do with the presence of $\sin x$ as well as $\tan x$. See the graphs here and here - all have a period of $2\pi$