In terms of x and y only, how would you write the following expression? $sin(tan^-1 x+cos^-1 y)$

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Answer 1 · 2016-12-06T01:19:24

$sin(tan^-1x+cos^-1y)$

Let $tan^-1x=A=>x=tanA$

Now

$sinA=tanAcosA=tanA/secA$

$=tanA/sqrt(1+tan^2A)$

$=x/sqrt(1+x^2)$

$cosA=1/secA=1/sqrt(1+tan^2A)=1/sqrt(1+x^2)$

Again let $cos^-1y=B=>y=cosB$

So $sinB=sqrt(1-cos^2B)=sqrt(1-y^2)$

Now the given expression

$=sin(tan^-1x+cos^-1y)$

$=sin(A+B)=sinAcosB+cosAsinB$

$=x/sqrt(1+x^2)xxy+1/sqrt(1+x^2)xxsqrt(1-y^2)$

$=(xy+sqrt(1-y^2))/sqrt(1+x^2)$

In terms of x and y only, how would you write the following expression? sin(tan^-1 x+cos^-1 y)sin(tan^-1 x+cos^-1 y)