Show that : cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A) = cos2A

Show that : 

cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A) = cos2A

Sol :

L.H.S. = cos(36 - A) cos(36 + A) + cos(54 + A) cos(54 - A)

           = 2/2 cos(36 - A) cos(36 + A) + 2/2 cos(54 + A) cos(54 - A)
       
          = 1/2{cos(36 -A +36 +A) + cos(36-A-36-A)} + 1/2{cos(54+A+54-A) + cos(54+A-54+A)}

         = 1/2 {cos72 + cos(-2A)} + 1/2 {cos108 + cos2A}

        = 1/2 {cos72 + cos108 +2cos2A}
   
        = 1/2 {2cos (72+108)/2 .cos(72-108)/2  + 2cos2A}

        = 1/2 {2 cos90 . cos16  + 2cos2A}

        = 1/2 . 2 cos2A            [cos90 = 0]

        = cos2A

        = R.H.S         PROVED




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