Tangent roots in algebraic equation?

Geometry Level 5

\[ \left( 1 + \dfrac{2x}{1-x^2} \right) \left( 1 + \dfrac{x^5 - 10x^3 + 5x}{5x^4 - 10x^2 + 1} \right) = 2 \]

The six roots of the equations above are \(x_1, x_2, \ldots, x_6 \). Given that

\[ x_1 = \tan \dfrac{b \pi }{c} , x_2 = \tan \dfrac{d \pi}e, x_3 = \tan \dfrac{f \pi}g , \\ x_4 = \tan \dfrac{h \pi}k , x_5 = \tan \dfrac{m\pi }n, x_6 = \tan \dfrac{z \pi }t, \]

where \(b,c,d,e,f,g,h,k,m,n,z\) are integers such that \(\gcd(b,c) = \gcd(d,e) = \gcd(f,g) = \gcd(h,k) = \gcd(m,n) = \gcd(z,t) = 1\).

If all the angles shown above are in the interval \( \left( -\dfrac \pi 2 , \dfrac\pi 2 \right) \), compute \(b+c+d+e+f+g+h+k+m+n+z+t\).


Inspiration.

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