If sin-1(1-x)+2sin-1(x)=π2, then x equals
0,-12
12,0
{0}
-1,0
Explanation for the correct option.
Step 1: Solve the given equation
Given that, sin-1(1-x)+2sin-1(x)=π2
⇒sin-1(1-x)=π2-2sin-1(x)⇒1-x=sinπ2-2sin-1(x)⇒1-x=cos2sin-1(x)….(i)
Put 2sin-1(x)=A⇒sin-1(x)=A2⇒sinA2=x
Step 2: Find the value of xUsing the identity cos2A=1-2sin2A⇒cosA=1-2sin2(A2)⇒cosA=1-2x2⇒A=cos-11-2x2From (i): ⇒1-x=coscos-11-2x2⇒1-x=1-2x2⇒2x2-x=0⇒x(2x-1)=0x=0,2x-1=0x=0,12
Hence, option B is correct.