Question

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

Answer 1

`"The Reqd. value="ysqrt(1-x^2)+xsqrt(1-y^2).`

Explanation:

Let, `sin^-1x=alpha, and,cos^-1y=beta`

We will consider only one case, namely, `0lex,yle1.`

Hence, `0 le alpha, beta le pi/2.`

Also, `sinalpha=x, cosbeta=y.`

Now, reqd. value`=cos(sin^-1x-cos^-1y)=cos(alpha-beta)`

`=cosalphacosbeta+sinalphasinbeta`

`=ycosalpha+xsinbeta.`

Now, `sinalpha=x rArr cosalpha=+-sqrt(1-sin^2alpha)=+-sqrt(1-x^2)`

But, `0 le alpha le pi/2 rArr cosalpha=+sqrt(1-x^2)`

Similarly, from `cosbeta=y", we get, "sinbeta=+sqrt(1-y^2).`Hence,

`"The Reqd. value="ysqrt(1-x^2)+xsqrt(1-y^2).`

We can deal with the other cases, like, `-1 le x le 0,` etc.,