SO I checked on my calculator and I was randomly searching values of: tan(89.99)=5729.577 tan(89.999)=57295.77 tan(89.9999)=57297.79 (all values approx) and so on..
But when I checked this via taylor series it's not coming to such a value. Plus it is similar with cosec(x) tending to 0 etc. So does this number hold some significance or its just a random number?
In terms of radians, for small $t>0$, $$\tan\left(\frac\pi 2-t\right)=\frac1{\tan t}\approx\frac1t.$$ If you are perverse enough to measure angles in degrees rather than radians, then $(90-t)^\circ=\frac\pi 2-\frac{\pi t}{180}$ so $$\tan(90-t)^\circ\approx\frac{180}{\pi t}.$$ So what does $180/\pi$ look like?