How do you simplify $sin ({cos}^{- 1} x)$ $sin(cos^-1x)$ ?

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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, $theta in [0,pi] rArr sinthetage0$

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

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

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