Verified Answer :

metchelle

So here is how we are going to prove that (2(tan x - cot x) / (tan^2 x - cot^2 x} = sin (2x). Follow it step by step:
LS  =   2(tanx−cotx)
        ------------------
           tan^2−cot^2x
     = 2(tanx−cotx)
    ----------------------
    (tanx−cotx)(tanx+cotx)
    =        2
      -------------------
        (tanx+cotx)
   =        2
      -----------------
       Sin2x+cos2x
       ----------------         sinxcosx
    =      2
      -------------
            1
      ------------
      sinxcosx
   = 2sinxcosx
   = sin2x
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