The usual arctangent addition formula works only when the arguments are in the proper range, namely: arctan⁡x+arctan⁡y=arctan⁡x+y1−xy, |xy|&lt;1 However for the other arguments we need to add some integer multiple of π.

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The usual arctangent addition formula works only when the arguments are in the proper range, namely:  arctan⁡x+arctan⁡y=arctan⁡x+y1−xy,   |xy|&amp;lt;1  However for the other arguments we need to add some integer multiple of π.

nick1337 · Accepted Answer

Doubling formula essentially ensures that we are working in interval (−π/4,π/4) instead of the usual (−π/2,π/2). The addition formula arctan⁡x+arctan⁡y=arctan⁡z,z=x+y1−xy is true if the value of left side lies in (−π/2,π/2) This should be obvious as by definition the right side lies in this interval. By using doubling formula you ensure that the arctan values lie in (−π/4,π/4) and the sum of two such values lies in (−π/2,π/2) Thus the resulting formula for addition of arctan values holds true for all the real values of x,y.

The usual arctangent addition formula works only when the arguments

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