How do you simplify $Cos(sin^-1 u + cos^-1 v)$ ?

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Answer 1 · 2016-04-13T11:38:28

$v*sqrt(1-u^2)-u*sqrt(1-v^2)$

$cos(sin^(-1)u+cos^(-1)v)=$

Making
$sin^(-1)u=alpha$ => $sin alpha=u$
$cos^(-1)v=beta$ => $cos beta=v$

And using the formula

$cos(alpha+beta)=cosalpha*cosbeta-sinalpha*sinbeta$

We get

$cos(alpha+beta)=sqrt(1-sin^2alpha)*cosbeta-sinalpha*sqrt(1-cos^2beta)$
$=sqrt(1-u^2)*v-u*sqrt(1-v^2)$
$=v*sqrt(1-u^2)-u*sqrt(1-v^2)$

How do you simplify Cos(sin^-1 u + cos^-1 v)Cos(sin^-1 u + cos^-1 v)?