Question

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

Answer 1

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

Explanation:

`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)`