Inverse Trigonometric Functions

Domains and principal ranges

$sin^{- 1} x : [- 1, 1] \to [- \frac{π}{2}, \frac{π}{2}],$ $cos^{- 1} x : [- 1, 1] \to [0, π]$
$tan^{- 1} x : R \to (- \frac{π}{2}, \frac{π}{2}),$ $cot^{- 1} x : R \to (0, π)$
$sec^{- 1} x : ∣ x ∣ \geq 1 \to [0, π] ∖ {\frac{π}{2}},$ $csc^{- 1} x : ∣ x ∣ \geq 1 \to [- \frac{π}{2}, \frac{π}{2}] ∖ {0}$

Complementary
$sin^{- 1} x + cos^{- 1} x = \frac{π}{2},$ $tan^{- 1} x + cot^{- 1} x = \frac{π}{2},$ $sec^{- 1} x + csc^{- 1} x = \frac{π}{2}$
Negative arguments
$sin^{- 1} (- x) = - sin^{- 1} x,$ $tan^{- 1} (- x) = - tan^{- 1} x,$ $cos^{- 1} (- x) = π - cos^{- 1} x,$ $cot^{- 1} (- x) = π - cot^{- 1} x$
Reciprocal
$csc^{- 1} x = sin^{- 1} \frac{1}{x},$ $sec^{- 1} x = cos^{- 1} \frac{1}{x},$ $cot^{- 1} x = tan^{- 1} \frac{1}{x} (x > 0)$
Compositions
$sin (sin^{- 1} x) = x (∣ x ∣ \leq 1);$ $sin^{- 1} (sin x) = x$ only for $x \in [- \frac{π}{2}, \frac{π}{2}]$

$tan^{- 1} x + tan^{- 1} y = tan^{- 1} \frac{x + y}{1 - x y} (x y < 1)$
If $x y > 1$ and $x, y > 0$ , add $π$ .
$tan^{- 1} x - tan^{- 1} y = tan^{- 1} \frac{x - y}{1 + x y} (x y > - 1)$
Double
$2 tan^{- 1} x = tan^{- 1} \frac{2 x}{1 - x ^{2}} = sin^{- 1} \frac{2 x}{1 + x ^{2}} = cos^{- 1} \frac{1 - x ^{2}}{1 + x ^{2}} (0 \leq x < 1)$
For $- 1 < x < 0$ the first two still hold; the $cos^{- 1}$ form becomes $- cos^{- 1} \frac{1 - x ^{2}}{1 + x ^{2}}$ .
Telescoping
$tan^{- 1} \frac{a _{k + 1} - a _{k}}{1 + a _{k} a _{k + 1}} = tan^{- 1} a_{k + 1} - tan^{- 1} a_{k}$
Write the general term this way to sum series of $tan^{- 1}$ terms.