Write the given expression in terms of x and y only cossin1x

Write the given expression in terms of x and y only. cos(sin^-1(x) - tan^-1(y)) ________ Find the exact value of the expression. sin(cos^-1(5/6) - tan^-1(1/3) ______

Solution

Cos(sin^-1x-tan^-1y)

This is of the form

Cos(A-B)=cosAcosB-sinAsinB

So we get

Cos(cos^-1x) cos(tan^-1y)-sin(cos^-1x)sin(tan^-1y)

cos(tan^-1y)

tan ^-1y=t

tan t=y

cos t=1/sqrt(1+y²)

Therefore

Cos(tan^-1x)=1/sqrt(1+y^2)

Sin(cos^-1x)

Cos^-1x=t

cos t=x

Sin t=sqrt(1-x²)

Therefore sin(cos^-1x)=sqrt(1-x²)

Sin(tan^-1y)

Tan^-1y=t

tan t=y

sin t=y/sqrt(1+y²)

So we get

x*1/(sqrt(1+y^2)+(sqrt(1-x^2)(y/sqrt(1+y^2))

(x+y sqrt(1-x^2))/(sqrt(1+y^2))