Prove the identity cos (pi/5)=[(sqrt 5) + 1]/4

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neela | High School Teacher | (Level 3) Valedictorian

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We know that  cos [2* (Pi/5)] = -cos [Pi - 2* (Pi/5)] =

= - cos [3* (Pi/2)]

Expanding both sides and putting cos(Pi/5) = x we get:

2x^2-1= -( 4x^3-3x) or

4x^3+2x^2-3x-1=0

(x+1)(4x^2-2x-1)=0

4x^2-2x-1 =0 gives: x= (2+sqrt20)/(2*4) or x= (2-sqrt20)/8 or x=-1,

But Pi/5 is an acute angle (equal t0 36 degree). So co 36 is a positive angle and x= (2+sqrt20)/8 or (1+sqrt5)/4 only valid.

So x = (1+sqrt5)/4 or

cos(Pi/5) = (1+sqrt5)/4

 

 

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