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Best Answer: You're wrong about what the arctan function is. arctan is that function that answers the question, What angle theta yields the tan (theta) = x? arctan 0 = 0 because the tan 0 = sin 0 / cos 0 = 0/1 = 0.
As metioned in Wikipedia's biography, Shanks used Machin's formula $$ \pi = 16\arctan(\frac15) - 4\arctan(\frac1{239}) $$. The standard way to use that (and the various Machin-like formulas found later) is to compute the arctangents using the power series. $$ \arctan x = x - \frac{x^3}3 + \frac{x^5}5 - \frac{x^7}7 + \frac{x^9}9 - \cdots $$. Getting $\arctan(\frac15)$ to 707 digits requires about 500 terms calculated to that precision.