How's this related to trig?

Calculus Level 5

75=x+x22!x33!x44!+x55!+x66!x77!x88!+...\large \dfrac{7}{5} = x + \dfrac{x^2}{2!} - \dfrac{x^3}{3!} - \dfrac{x^4}{4!} + \dfrac{x^5 }{5!} + \dfrac{x^6}{6!} - \dfrac{x^7}{7!} - \dfrac{x^8}{8!} + ...

If the value of x x that satisfies the above equation can be expressed in the form 2arctan(a+bc)+2πk 2 \arctan \left( \dfrac{-a + \sqrt{b}}{c}\right) + 2\pi k for integer kk, where a a and cc are positive coprime integers and b b is square-free, find a+b+c a + b + c .

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