Melody,
arctan x is a multivalued function, it has an infinite number of branches both above and below the one shown by desmos, desmos just shows the principal range.
Care must be taken when giving the values for x.
sec(theta) is negative so theta is in the second or third quadrant, so
\(\displaystyle \tan^{-1}x= \pi \pm \pi/6\) ,
meaning that
\(\displaystyle x = \tan(\pi \pm \pi/6)\), angle \(\displaystyle \pm2k\pi\) for the general solution.
It's tempting to now say that \(\displaystyle x = \pm1/\sqrt{3}\), but this would be wrong, (without some qualification).
\(\displaystyle x = 1/\sqrt{3}\), for example, includes the possibility that the angle is \(\displaystyle \pi/6\), which is not a solution of the original equation.
-Bertie
