SOLUTION: Prove that : cos(72°) =(1/2)sqrt(4sin²(36)-1)

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Question 1191002: Prove that : cos(72°) =(1/2)sqrt(4sin²(36)-1)
Answer by MathLover1(20849) About Me  (Show Source):
You can put this solution on YOUR website!

Prove that : cos%2872%29+=%281%2F2%29sqrt%284sin%5E2%2836%29-1%29
solve left side:
cos+%2872%29 .....use identity cos%28+72%29++=cos+%2890+-+18%29°
use identity
cos%28a-b%29=sin%28a%29+sin%28b%29+%2B+cos%28a%29+cos%28b%29........where a=90° and b=18°

cos+%2890+-+18%29+=sin%2890%29+sin%2818%29+%2B+cos%2890%29+cos%2818%29
+cos+%2890+-+18%29+=1%2A+sin%2818%29+%2B+0%2Acos%2818%29
cos+%2890+-+18%29+=+sin%2818%29+

cos+%2890+-+18%29+=%281%2F4%29+%28sqrt%285%29+-+1%29


solve right side:
%281%2F2%29sqrt%284sin%5E2%2836%29-1%29.....sin%5E2%2836%29=5%2F8+-+sqrt%285%29%2F8
=%281%2F2%29sqrt%284+%285%2F8+-+sqrt%285%29%2F8%29+-+1%29
=%281%2F2%29sqrt%28%281%2F2%29+%283+-+sqrt%285%29%29%29
=%281%2F2%29%28sqrt%285%29%2F2+-+1%2F2%29
=%281%2F4%29%28sqrt%285%29+-+1%29

so, both sides have same answer
then
cos%2872%29+=%281%2F2%29sqrt%284sin%5E2%2836%29-1%29
%281%2F4%29%28sqrt%285%29+-+1%29=%281%2F4%29%28sqrt%285%29+-+1%29 ->proven