Question

How do you prove `sin(cos^-1x)=sqrt(1-x^2)`?

Answer 1

`sin(cos^-1 x) =sin(sin^-1sqrt(1-x^2))=sqrt(1-x^2)` (proved)

Explanation:

Let `cos^-1x=theta :. cos theta= x`. We know `cos theta=a/h =x/1` where a(=x),adjacent side in a right angled triangle , and h(=1),is the hypotenuse, then by pythagorus theorm ,opposite side (o). `o=sqrt(1-x^2) :.sin theta= o/h=sqrt(1-x^2)/1 :. theta=sin^-1 sqrt(1-x^2)`

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