Inverse Trigonometric Functions – Form f(f⁻¹(x))

Inverse Trigonometric Functions – Form f(f⁻¹(x))

Function of the form f(f⁻¹(x)):

where f(x) is Trigonometric Function

(i)Consider function f(x) = sin(sin⁻¹x).

Domain of the function is [-1, 1]

Also, sin (sin ⁻¹(x)) = x

sin(sin⁻¹x) = x for all x ϵ [-1, 1]

(ii)Consider function f(x) = cos(cos⁻¹x).

Domain of the function is [-1, 1]

Also, cos(cos⁻¹x) = x

cos(cos⁻¹x) = x for all x ϵ [-1, 1]

Graph of y = sin(sin⁻¹x) and y = cos(cos⁻¹x)

(iii)Consider function f(x) = tan(tan⁻¹x) = x, for all x ϵ R

(iv)Consider function f(x) = cot(cot⁻¹x) = x, for all x ϵ R

Graph of y = Tan(tan⁻¹x) and y = cot(cot⁻¹x)

(v)Consider function f(x) = cosec(cosec⁻¹x) = x, for all x ϵ (-∞, -1] $$\cup$$ [1, ∞)

(vi)Consider function f(x) = sec(sec⁻¹x) = x, for all x ϵ (-∞, -1] $$\cup$$ [1, ∞)

Graph of y = cosec(cosec⁻¹x) and y = sec(sec⁻¹x)