It isn't clear what you mean by "underroot"; I am not aware of a mathematical function by that name.
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(-x+tanx)'=-1+(1/cos2x)
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d/dx(1+tanx)=0+sec2x=sec2x
I assume you mean (tanx+1)^2 In which case, (tanx+1)^2=tan2x+2tanx+1
(tanx+cotx)/tanx=(tanx/tanx) + (cotx/tanx) = 1 + (cosx/sinx)/(sinx/cosx)=1 + cos2x/sin2x = 1+cot2x= csc2x This is a pythagorean identity.