The truth behind the signature arctangent question is simply that the arctangent addition formula, in direct resemblance to the tangent addition formula, is directly related to the fact that 1/3 + 1/7 equals 7/21 + 3/21 and the fact that 7 + 3 = 10, resulting in a final solution of 10/21 just in the numerator. In the denominator, however, is 1 - (1/3)(1/7), which results in 1 - 1/21, which in return results in (21-1)/21 equals 20/21. If the numerator is 10/21 and the denominator is 20/21, the ratio is exactly identical to the ratio with numerator 10 and denominator 20. Furthermore, 10/20 is equivalent to the simplified expression 1/2.
Related Question: Why can Pi be represented by arctan(1)+arctan(2)+arctan(3)?
The arctangent of one plus the arctangent of two is equivalent to the arctangent of (1+2)/(1-1*2), which is the arctangent of 3/-1 equals -3. With two and three, the formula becomes (2+3)/(1-2*3), which becomes 5/-5 and thus becomes -1. For one and three, the formula is (1+3)/(1-1*3), which equates to 4/-2 equals -2. Multiplying the arctangents of the negatives by the multiplicative inverses effectively cancels out the numerators, as in x+(-x) equals x-x equals 0 (zero). Even though the arctangent of zero is, in fact, zero, the tangent function is notable for having a period of π, and with the positive arctangents, arctan(1)+arctan(2)+arctan(3) can, in fact, equal π.
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