How do you find the exact value of $sin(cos^-1(sqrt5/5))$ ?

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Answer 1 · 2018-02-23T05:43:24

$sin(cos^-1(sqrt(5)/5))=(2sqrt(5))/5$

Let $cos^-1(sqrt(5)/5)=A$ then $cosA=sqrt(5)/5$

and $sinA=sqrt(1-cos^2A)=sqrt(1-(sqrt(5)/5)^2)=(2sqrt(5))/5$
$rarrA=sin^-1((2sqrt(5))/5)$

Now, $sin(cos^-1(sqrt(5)/5))=sin(sin^-1((2sqrt(5))/5))=(2sqrt(5))/5$

How do you find the exact value of sin(cos^-1(sqrt5/5))sin(cos^-1(sqrt5/5))?