Inverse Tangent Identities
The inverse tangent function obeys a family of addition and doubling identities that let you combine two arctangents into one. They come straight from the tangent addition formula, but they carry conditions you must respect.
The core addition identity
Start from tan(A+B)=1−tanAtanBtanA+tanB. Put A=tan−1x and B=tan−1y, so tanA=x and tanB=y. Then
tan−1x+tan−1y=tan−1(1−xyx+y),xy<1.
The restriction xy<1 keeps the combined angle inside the principal range (−π/2,π/2).
If xy>1 the raw formula lands in the wrong branch, so you must correct it:
tan−1x+tan−1y=π+tan−1(1−xyx+y) (x,y>0),
and −π+tan−1(⋅) when x,y<0. Ignoring this is the classic exam slip.
Subtraction
Replacing y with −y gives
tan−1x−tan−1y=tan−1(1+xyx−y),xy>−1.
The doubling identity
Set y=x in the addition formula:
2tan−1x=tan−1(1−x22x),−1<x<1.
The same angle can also be rewritten through sine and cosine, which is handy in integration and in proofs -- but each alternate form only matches 2tan−1x on part of its domain, so the two forms carry different conditions:
2tan−1x=sin−1(1+x22x),−1≤x≤1,
2tan−1x=cos−1(1+x21−x2),x≥0.
The cos−1 form needs x≥0 -- it fails for negative x. Check x=−1: 2tan−1(−1)=2(−4π)=−2π, but cos−1(1+11−1)=cos−1(0)=2π, the wrong sign entirely. The sin−1 form has no such restriction because sin−1 (unlike cos−1) can return a negative angle.
The complementary identity
For every real x,
tan−1x+cot−1x=2π.
This holds without restriction because tan−1 and cot−1 of the same value are complementary angles. …