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) 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

Answer by Edwin McCravy(20055) About Me  (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,

AB%2FAD=AC%2FAB, or

x%2F1=%281%2Bx%29%2Fx

x%5E2=1%2Bx

x%5E2-x-10

Solving for x by the quadratic formula:

x+=+%281+%2B+sqrt%285%29%29%2F2+

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

By the law of cosines on ΔDAB,

BD%5E2=AB%5E2%2BAD%5E2-2%2AAB%2AAD%2Acos%28A%29

x%5E2=x%5E2%2B1%5E2-2%2Ax%2A1%2Acos%2872%5Eo%29 

2x%2Acos%2872%5Eo%29=1

cos%2872%5Eo%29=1%2F%282x%29=1%2F%282%28%281%2Bsqrt%285%29%29%2F2%29%29

Simplifying and rationalizing the denominator,

cos%2872%5Eo%29=%28sqrt%285%29-1%29%2F4%29%29

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

By the law of sines on ΔDAB,

AD%2Fsin%28%22%22%3CABD%29=BD%2Fsin%28A%29

1%2Fsin%2836%5Eo%29=x%2Fsin%2872%5Eo%29

sin%2872%5Eo%29=x%2Asin%2836%5Eo%29

sqrt%281-cos%5E2%2872%5Eo%29%29=%28%281+%2B+sqrt%285%29%29%2F2%29sin%2836%5Eo%29

Multiply both sides by 2

2sqrt%281-cos%5E2%2872%5Eo%29%29=%281+%2B+sqrt%285%29%29sin%2836%5Eo%29

Square both sides:

4%281-cos%5E2%2872%5Eo%29%29=%281+%2B+sqrt%285%29%29%5E2%2Asin%5E2%2836%5Eo%29

4-4cos%5E2%2872%5Eo%29=%281+%2B+2sqrt%285%29%2B5%29%2Asin%5E2%2836%5Eo%29

4-4%28%28sqrt%285%29-1%29%2F4%29%5E2=%286+%2B+2sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

4-4%285-2sqrt%285%29%2B1%29%2F16=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

4-%286-2sqrt%285%29%29%2F4=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

4-2%283-sqrt%285%29%29%2F4=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

4-%283-sqrt%285%29%29%2F2=2%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

Multiply through by 2

8-%283-sqrt%285%29%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

8-3%2Bsqrt%285%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

5%2Bsqrt%285%29=4%283+%2B+sqrt%285%29%29%2Asin%5E2%2836%5Eo%29

%285%2Bsqrt%285%29%29%2F%284%283+%2B+sqrt%285%29%29%29=sin%5E2%2836%5Eo%29

Rationalize the denominator:

sin%5E2%2836%5Eo%29=%285-sqrt%285%29%29%2F8

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

cos%2872%5Eo%29+=%281%2F2%29sqrt%284sin%5E2%2836%5Eo%29-1%29

%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%284%28%285-sqrt%285%29%29%2F8%29-1%29

%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%28%285-sqrt%285%29%29%2F2%29-1%29

%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%28%285-sqrt%285%29%29%2F2%29-2%2F2%29

%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%285-sqrt%285%29-2%29%2F2%29

%28sqrt%285%29-1%29%2F4+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29

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

sqrt%28%28+%28sqrt%285%29-1%29%2F4%29%5E2+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29



sqrt%28%28+%286-2sqrt%285%29%29%2F16%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29

sqrt%28%28+2%283-sqrt%285%29%29%2F16%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29

sqrt%28+%283-sqrt%285%29%29%2F8+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29

sqrt%28+%283-sqrt%285%29%29%2F%284%2A2%29+%29+=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29



%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29=%281%2F2%29sqrt%28%283-sqrt%285%29%29%2F2%29

Edwin