How do you find $sin(cos^-1x)$ ?

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Answer 1 · 2016-09-24T08:52:14

$sqrt(1-x^2)$ .

Let $a =cos^(-1)x in [0, pi]$ . for the principal value. Then cos a =x, for

both positive and negative x,..

The given expression is

$sin a=sqrt(1-cos^2a)= sqrt(1-x^2)$ , for $a in [0, pi]$ ..

If the principal value convention is relaxed and x is negative, a could

be in $Q_3$ wherein both sine and cosine are negative. And then,

$sin a = -sqrt(1-x^2)$ ,

.

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