How do you simplify #sin(cos^-1x)#?

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Answer 1 · 2016-09-25T09:50:19

#:. sin(cos^-1x)=sqrt(1-x^2)#.

Let #cos^-1x=theta, |x|le1," so that, "sin(cos^-1x)=sintheta.#

#"By the Defn. of "cos^-1" fun.," cos^-1x=thetarArrcostheta=x#,

where, #theta in [0,pi].#

#" Reqd. Value="sintheta=+-sqrt(1-cos^2theta)=+-sqrt(1-x^2).#

But, #theta in [0,pi] rArr sinthetage0#

#:." Reqd. Value="+sqrt(1-x^2).#

#:. sin(cos^-1x)=sqrt(1-x^2)#.