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)#