Example-1 :-  Find the principal value of sin⁻¹(1/√2).

   Let sin⁻¹(1/√2) = θ.
       sin θ = 1/√2
       sin θ = sin(π/4)  {Range = [-π/2, π/2]}
           θ = π/4
   The principal value of sin⁻¹(1/√2) = π/4
    

Example-2 :-  Find the principal value of cot⁻¹(-1/√3).

   Let cot⁻¹(-1/√3) = θ.
       cot θ = -1/√3
       cot θ = -cot(π/3)  {Range = [0, π]}
       cot θ = cot(π - π/3)
       cot θ = cot(2π/3)
           θ = 2π/3
   The principal value of cot⁻¹(-1/√3) = 2π/3

Example-3 :-  Show that
(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x
(ii) sin⁻¹(2x√1 - x²) = 2cos⁻¹x.

(i) sin⁻¹(2x√1 - x²) = 2sin⁻¹x
   Let x = sin θ, then θ = sin⁻¹x 
   L.H.S
   sin⁻¹(2x√1 - x²)
 = sin⁻¹(2 sin θ √1 - sin² θ)        ∴[√1 - sin² θ = cos θ]
 = sin⁻¹(2 sin θ cos θ)    ∴[sin 2θ = 2 sin θ cos θ]
 = sin⁻¹(sin 2θ)
 = 2θ 
 = 2 sin⁻¹x  
 = R.H.S

(ii)sin⁻¹(2x√1 - x²) = 2cos⁻¹x.
   Let x = cos θ, then θ = cos⁻¹x 
   L.H.S
   sin⁻¹(2x √1 - x²)
 = sin⁻¹(2 sin θ √1 - sin² θ)        ∴[√1 - sin² θ = cos θ]
 = sin⁻¹(2 sin θ cos θ)    ∴[sin 2θ = 2 sin θ cos θ]
 = sin⁻¹(sin 2θ)
 = 2θ 
 = 2 cos⁻¹x
 = R.H.S 
    

Example-4 :-  Show that

Example-5 :-  Express , -3π/2 < x < π/2 in the simplest form.

Example-6 :-  Write , x > 1 in the simplest form.

Example-7 :-  Prove that

Example-8 :-  Find the value of cos(sec⁻¹x + cosec⁻¹x), | x | ≥ 1.

   cos(sec⁻¹x + cosec⁻¹x)   
 ∴[sec⁻¹x + cosec⁻¹x = π/2]
 = cos(π/2)
 = 0