Question

prove that cot 2A-cot2B=(cos2A-cos2B)/(sin2 A.sin2B)=cosec2 A- cosec2 B

Answer 1

1.cot²a-cot²b = cos²a-cos²b/sin²a.sin²b 
L.H.S;
        = cos²a÷sin²a - cos²b÷sin²b
       =cos²a .sin²b -cos²b.sin²a ÷ sin²a.sin²b
      =cos²a(1-cos²b) - cos²b {1-cos²a} ÷sin²a .sin²b
      = cos²a - cos²a cos²b -cos²b +cos²a cos²b ÷sin²a.sin²b
      = cos²a-cos²b ÷sin²a.sin²b { R.H.S}
2.  =cos²a.sin²b-cos²b.sin²a /sin²a.sin²b
     ={1-sin²a}sin²b - (1-sin²b)sin²a /sin²a.sin²b
     = sin²b-sin²a/sin²a.sin²b
     = sin²b/sin²a.sin²b  - sin²a/sin²a.sin²b
     = 1/sin²a - 1/sin²b
     =cosec²a-cosec²b     
hence proved...