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

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Question 1191003: Prove that : cos(72°) =(1/2)sqrt(4sin²(36°)-1)
Found 2 solutions by MathLover1, Edwin McCravy:
Answer by MathLover1(20849)   (Show Source): You can put this solution on YOUR website!
Prove that :
solve left side:
.....use identity °
use identity
........where ° and °








solve right side:
.....
=
=
=
=

so, both sides have same answer
then

->proven

Answer by Edwin McCravy(20055)   (Show Source): You can put this solution on YOUR website!
She used special exact values for certain trig ratios for certain angles.
I didn't use them. I begin by drawing an isosceles triangle with a vertex
angle of 36o and base angles 72o each.



Now draw the internal bisector BD of the 72o angle B on the right 
into two 36o angles. Triangle ABD is isosceles since its base
angles are both 36o. Then the linear pair of angles at D are 72o and 108o. 



ΔDAB ∽ ΔABC because their internal angles are congruent.

Let AD = 1 and AB = x. Then by isosceles triangles

AB = DB = CD = x

By similar triangles ΔDAB and ΔABC,

, or







Solving for x by the quadratic formula:



We must use the plus sign since the minus sign
gives a negative value for x.

By the law of cosines on ΔDAB,



 





Simplifying and rationalizing the denominator,



I will use that for the left side of the identity.

By the law of sines on ΔDAB,









Multiply both sides by 2



Square both sides:















Multiply through by 2









Rationalize the denominator:



I will use that on the right side of the original identity
which is to be proved:













Since the left side is positive, it is the positive square 
root of its square:

















Edwin
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