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

1 Answer
Oct 14, 2016

#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)