4.10 Properties of Inverse Trigonometric Functions

. Properties of Inverse Trigonometric Functions In this section, we investigate some properties of inverse trigonometric functions. The properties to be discussed are valid within the principal value branches of the corresponding inverse trigonometric functions and where they are defined. Property-I (i) sin (sin ) θ , if θ ∈− (ii) cos (cos ) θ , if θ ∈ [ , ] (iii) tan (tan ) θ , if θ ∈− .

(iv) cosec cosec ( , if θ ∈−   { } \ (v) sec (sec ) θ , if θ ∈   [ , ] \ . (vi) cot (cot ) θ , if θ ∈ ( , Proof All the above results follow from the definition of the respective inverse functions. For instance, (i) let sin θ = x ; θ ∈− Now, sin θ = x gives θ = sin x , by definition of inverse sine function. Thus, sin − ( ) = θ .

Property-II (i) sin sin − ) = x x , if x ∈− , ] . (ii) cos cos − ) = x x, if x ∈− , ] (iii) tan tan − ) = x x , if x ∈  (iv) cosec cosec − ) = x x, if x ∈  \ ( , ) (v) sec sec − ) = x x , if x ∈  \ ( , ) (vi) cot cot − ) = x x , if x ∈  Proof (i) For x ∈− ] , sin - x is well defined. Let sin − x θ . Then, by definition θ ∈− and sin θ = x Thus, sin θ = x implies sin sin −