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

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

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Answer 1 · 2016-04-13T11:38:28

$v \cdot \sqrt{1 - u^{2}} - u \cdot \sqrt{1 - v^{2}}$ $v*sqrt(1-u^2)-u*sqrt(1-v^2)$

Explanation:

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

Making
${sin}^{- 1} u = α$ $sin^(-1)u=alpha$ => $sin α = u$ $sin alpha=u$
${cos}^{- 1} v = β$ $cos^(-1)v=beta$ => $cos β = v$ $cos beta=v$

And using the formula

$cos (α + β) = cos α \cdot cos β - sin α \cdot sin β$ $cos(alpha+beta)=cosalpha*cosbeta-sinalpha*sinbeta$

We get

$cos (α + β) = \sqrt{1 - {sin}^{2} α} \cdot cos β - sin α \cdot \sqrt{1 - {cos}^{2} β}$ $cos(alpha+beta)=sqrt(1-sin^2alpha)*cosbeta-sinalpha*sqrt(1-cos^2beta)$
$= \sqrt{1 - u^{2}} \cdot v - u \cdot \sqrt{1 - v^{2}}$ $=sqrt(1-u^2)*v-u*sqrt(1-v^2)$
$= v \cdot \sqrt{1 - u^{2}} - u \cdot \sqrt{1 - v^{2}}$ $=v*sqrt(1-u^2)-u*sqrt(1-v^2)$

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

1 Answer

Explanation:

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