Question

find the value of cos18

Answer 1

          5 * 18 = 90
       so,  3 * 18 = 90 - 2 * 18
        
   Cos (3 * 18) = Cos (90 - 2 * 18)
   4 Cos³ 18 - 3 Cos 18  = Sin 2 * 18  = 2 Sin 18  Cos 18

As Cos 18  ≠ 0,  cancelling the term
 
    4 Cos² 18 - 3 = 2 Sin 18
    4 - 4 Sin²18 - 3 = 2 Sin 18
    
    4 Sin² 18 + 2 Sin 18 - 1 = 0

         Sin 18 =  (-2 +-√(4+16 )/8 = (√5 - 1) / 4    as sin 18 is positive.
        
      Cos 18 =   √(16 - 5 -1 + 2√5) / 4 = √(10+2√5)  / 4