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Define
f(x)=12tan(x2)+14tan(x4)+18tan(x8)+⋯ .f(x) = \dfrac{1}{2} \tan\left(\dfrac{x}{2}\right) + \dfrac{1}{4} \tan\left(\dfrac{x}{4}\right) + \dfrac{1}{8} \tan\left(\frac{x}{8}\right) +\cdots.f(x)=21tan(2x)+41tan(4x)+81tan(8x)+⋯.
Then f(3π8)f\left(\dfrac{3\pi}{8}\right) f(83π) can be written as abπc+d−e \dfrac{a}{b\pi^{c}} + d - \sqrt{e}bπca+d−e
If ab\dfrac{a}{b}ba is in simplest form i.e. gcd(a,b)=1\gcd(a,b)=1gcd(a,b)=1, and a,b,c,d,e∈Na,b,c,d,e\in\mathbb Na,b,c,d,e∈N, find the value of a+b+c+d+ea + b + c + d + ea+b+c+d+e.
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