What input did you give Wolfram|Alpha, exactly? If I give

tan x-sec x=sqrt((1-sin x)/(1+sin x))

the output is more complicated than what you said. It also gives

2 pi n-pi<x<1/2 (4 pi n-pi),   n element Z

which covers the intervals between pi and 3/2pi. It seems that W|A misses just the solutions x=pi+2n Pi. Mathematica does not miss them. W|A also says "Integer solutions", which does not make much sense. As Wolfram Language plaintex input it gives

Reduce[-Sec[x] + Tan[x] == 
  Sqrt[(1 - Sin[x])/(1 + Sin[x])], x, Integers]

which seems to assume that you are searching for integer solutions only.