Aleksey,
There is a discontinuity there:
In[20]:= Limit[ArcSin[Cos[x]]/ArcCos[Abs[Sin[Abs[x]]]], x -> Pi/2,
Direction -> "FromAbove"]
Out[20]= -1
In[21]:= Limit[ArcSin[Cos[x]]/ArcCos[Abs[Sin[Abs[x]]]], x -> Pi/2,
Direction -> "FromBelow"]
Out[21]= 1
You can evaluate the expression with arbitrary accuracy by specifying the accuracy. Michael is correct that as soon as you add the 0.0000001 to Pi/2 you use machine precision. The syntax is this:
ArcSin[Cos[x]]/ArcCos[Abs[Sin[Abs[x]]]] /.
x -> Pi/2 + .000000000000000000000000000000001`50
to get
-1.0000000000000000000000000000000000000000000000000
(Note: in this case I specified 50 digits of precision. You can go out to whatever you want.)