Prove:

(2(tan x - cot x) / (tan^2 x - cot^2 x} = sin (2x)

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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