Hmmm,

I think the gap is due to the method how Mathematica produces plots, gathering values at different positions and refining and so. The other question: Tan[ x ] = 1 means x = Pi / 4 , and for this x both constituents of your functions have the same values as well as their derivatives:

In[17]:= y1 = 1 - Tan[x]/2;
y2 = Cot[x]/2;
y1 /. x -> Pi/4
y2 /. x -> Pi/4

Out[19]= 1/2

Out[20]= 1/2



  (\!\(
   \*SubscriptBox[\(\[PartialD]\), \(x\)]\ y1\)) /. x -> Pi/4
   (\!\(
   \*SubscriptBox[\(\[PartialD]\), \(x\)]\ y2\)) /. x -> Pi/4

   -1

   -1

But the second derivatives are different, meaning the derivative of the composite function is not smooth. exactly as you see it. So it seems to me nothing is weird.

Look at

Plot[{y1, y2}, {x, 0, 1.5}, Plot[{y1, y2}, {x, 0, 1.5}, PlotStyle -> {Red, Blue}] 

Plot[Evaluate[{D[y1, x], D [y2, x]}], {x, 0, 1.5},  PlotStyle -> {Red, Blue}] 

Plot[Evaluate[{D[y1, x, x] , D[ y2, x, x]}], {x, 0, 1.5},  PlotStyle -> {Red, Blue}]