Question

Simplify sin^-1[(x+(√1-x^2))/√2]?

Answer 1

`sin^-1((x+sqrt(1-x^2))/sqrt2)=sin^(-1)x+pi/4`

Explanation:

`sin^-1((x+sqrt(1-x^2))/sqrt2)`

Let `x=sinalpha` i.e. `alpha=sin^(-1)x` and then

`((x+sqrt(1-x^2))/sqrt2)`

= `(sinalpha+sqrt(1-sin^2alpha))/sqrt2`

= `(sinalpha+cosalpha)/sqrt2`

= `(sinalpha*1/sqrt2+cosalpha*1/sqrt2)`

= `sinalpha*cos(pi/4)+cosalpha*sin(pi/4)`

= `sin(alpha+pi/4)`

Hence `sin^-1((x+sqrt(1-x^2))/sqrt2)=alpha+pi/4=sin^(-1)x+pi/4`