Question 9727
 First of all, alway remember to put parentheses to group related terms,
 as P= (x+1)/(x-1) and Q=(x-1)/(x+1).

 Note, P = 1 + [2/(x-1)] and  Q = 1/P = 1 + [-2/(x+1)] 
 So,P-Q = 2[1/(x-1) + 1/(x+1)] = 2[2x/(x^2-1)]
   =  4x/(x^2-1) 
 and PQ = ??(for you)

 Since P^2+Q^2-PQ  = (P-Q)^2 +PQ (why ?)
 We have P^2+Q^2-PQ  = [4x/(x^2-1)]^2 + 1
  = 1 + 16x^2 /(x^2-1)^2 

 Of course ,you can use direct substitution and staightford computation
 to get the same asnwer. But, the above way it faster and can avoid some
 mistakes.
 
 I won't give further explanations for this level of questions.

 Kenny