I want to show that x=cos⁡π11 is a solution of equation :8x2−4x+1x−4=41−x2

We have to prove that for x=cos⁡π11 we have:4x(2x2−1)−(4x2−1)=4x1−x2or:4cos⁡π11cos⁡2π11−sin⁡3π11sin⁡π11=4cos⁡π11sin⁡π22that is equivalent to:sin⁡4π11−sin⁡3π11=2sin⁡2π11sin⁡π22or to:2cos⁡7π22sin⁡π22=2sin⁡2π11sin⁡π22that is trivial since cos⁡7π22=sin⁡4π22=sin⁡2π11

"I want to show that x=cos⁡π11 is a..." solved and explained

Blaine Jimenez

Answered question

2022-04-07

I want to show that

x = \cos \frac{π}{11}

is a solution of equation :

8 x^{2} - 4 x + \frac{1}{x} - 4 = 4 \sqrt{\frac{1 - x}{2}}

Answer & Explanation

bondsk384r

Beginner2022-04-08Added 12 answers

We have to prove that for

x = \cos \frac{π}{11}

we have:

4 x (2 x^{2} - 1) - (4 x^{2} - 1) = 4 x \sqrt{\frac{1 - x}{2}}

or:

4 \cos \frac{π}{11} \cos \frac{2 π}{11} - \frac{\sin \frac{3 π}{11}}{\sin \frac{π}{11}} = 4 \cos \frac{π}{11} \sin \frac{π}{22}

that is equivalent to:

\sin \frac{4 π}{11} - \sin \frac{3 π}{11} = 2 \sin \frac{2 π}{11} \sin \frac{π}{22}

or to:

2 \cos \frac{7 π}{22} \sin \frac{π}{22} = 2 \sin \frac{2 π}{11} \sin \frac{π}{22}

that is trivial since

\cos \frac{7 π}{22} = \sin \frac{4 π}{22} = \sin \frac{2 π}{11}

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