Question

sin-1(1-x)-2sin-1x= pi/2 ? solve for x?please help me out.

Answer 1

let  sin^-1  (1-x) = A  ,   So  Sin A  = 1-x
let  Sin^-1 x = B,     So Sin B = x 

Given   A - 2 * B = π/2
            A  - π/2  = 2B
           Cos (A - π/2)  = Cos (2 B)
           Sin A =  1 - 2 Sin² B 
           1 - x   = 1 - 2 x² 
           2 x² - x = 0
           x = 0 or   x = 1/2
       

Answer 2

sin-¹ (1-x) -2sin-¹ x = π/2

sin-¹ (1-x) = π/2 + 2sin-¹x

( 1-x) = sin(π/2 + 2sin-¹x )

(1 -x) = cos( 2sin-¹x)

let 2sin-¹x = t

x = sint/2
cost = 1 -2sin²t/2 =(1 -2x²)

so, t =cos-¹(1-2x²)
then,

( 1 -x ) = cos{cos-¹(1-2x²)}
1 -x = 1 -2x² when -1≤ 1-2x² ≤ 1

x= 0, 1/2 and -2≤ -2x² ≤ 0

x =0, 1/2 and 0 ≤ 2x² ≤2

x =0 , 1/2 and 0 ≤ x ≤ 1

so, x =0 , 1/2