Question

Solve the equation for x:sin⁻¹x+sin⁻¹(1−x)=cos⁻¹x

Answer 1

We have,

sin⁻¹x+sin⁻¹(1−x)=cos⁻¹x

`sin^−1 x -cos^−1 x=-sin^−1 (1−x)`

`sin^−1 x -cos^−1 x=sin^−1 (x-1) ......................(1)   [because sin^(-1)(-x)=-sin^-1x]`

`Put sin^-1 x=theta and cos^-1 x= phi`

`sin theta=x and cos phi=x`

`then cos theta=sqrt(1-sin^2theta) and sin phi=sqrt(1-cos^2 phi)`

`cos theta=sqrt(1-x^2) and sin phi =sqrt(1-x^2)`

Applying the formula:

`sin(theta-phi)=sin theta cos phi-cos theta sin phi` , we get

`sin(theta-phi)=x.x-sqrt(1-x^2)sqrt(1-x^2)`

`sin(theta-phi)=x^2-(1-x^2)`

`sin(theta-phi)=x^2-1+x^2`

`sin(theta-phi)=2x^2-1`

`(theta-phi)=sin^-1(2x^2-1)`

`sin^-1x - cos^-1 x=sin^-1(2x^2-1).............(2)`

From (1)  and  (2), we get 

`sin^-1 (2x^2-1)= sin^-1 (x-1)`

`2x^2-x=0`

`x(2x-1)=0`

`x=0 or 2x-1=0`

`x=0 or x=1/2`