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prove that (1-cosA+cosB-cos(A+B))/(1+cosA-cosB-cos(A+B)=Tan(A/2)*(cotB/2) Plz help me out sir.

prove that
(1-cosA+cosB-cos(A+B))/(1+cosA-cosB-cos(A+B)=Tan(A/2)*(cotB/2)
Plz help me out sir.
 

Grade:12

1 Answers

Himanshu
103 Points
5 years ago
L.H.S. = (1-cosA+cosB-cos(A+B))/(1+cosA-cosB-cos(A+B)
  = [(1-cos(A+B)) – (cosA-cosB)] / [(1-cos(A+B) + (cosA-cosB)]
  = [2.sin2((A+B)/2) + 2.sin((A+B)/2).sin((A-B)/2)] / 
                  [2.sin2((A+B)/2) - 2.sin((A+B)/2).sin((A-B)/2)]
  = 2.sin((A+B)/2) [sin((A+B)/2) + sin((A-B)/2)] / 
                  2.sin((A+B)/2) [sin((A+B)/2) - sin((A-B)/2)]
  = [sin((A+B)/2) + sin((A-B)/2)] / [sin((A+B)/2) - sin((A-B)/2)]
  = 2.sin(A/2).cos(B/2) / 2.sin(B/2).cos(A/2)
  = tan(A/2).(cotB/2)
  = R.H.S.
Hence, proved.

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