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Showing content from https://dlmf.nist.gov/4.24 below:

Further Properties ‣ Trigonometric Functions ‣ Chapter 4 Elementary Functions

§4.24 Inverse Trigonometric Functions: Further Properties Contents
  1. §4.24(i) Power Series
  2. §4.24(ii) Derivatives
  3. §4.24(iii) Addition Formulas
§4.24(i) Power Series

which requires z ( = x + i ⁢ y ) to lie between the two rectangular hyperbolas given by

§4.24(ii) Derivatives 4.24.7 d d z ⁡ arcsin ⁡ z = ( 1 − z2 ) − 1 / 2 , 4.24.8 d d z ⁡ arccos ⁡ z = − ( 1 − z2 ) − 1 / 2 , 4.24.9 d d z ⁡ arctan ⁡ z = 1 1 + z2 . 4.24.10 d d z ⁡ arccsc ⁡ z = ∓ 1 z ⁢ ( z2 − 1 ) 1 / 2 , ℜ ⁡ z ≷ 0 . 4.24.11 d d z ⁡ arcsec ⁡ z = ± 1 z ⁢ ( z2 − 1 ) 1 / 2 , ℜ ⁡ z ≷ 0 . 4.24.12 d d z ⁡ arccot ⁡ z = − 1 1 + z2 . §4.24(iii) Addition Formulas

The above equations are interpreted in the sense that every value of the left-hand side is a value of the right-hand side and vice versa. All square roots have either possible value.

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