Question

Find `cos^(-1)x-cos^(-1)y`, if `x=1/4` and `y=2/3`?

Answer 1

`cos^-1(1/12(2+5sqrt(3)))`

Explanation:

Let `z=cos^-1x-cos^-1 y` then

`cosz=cos(cos^-1x-cos^-1 y)`

but `cos(a-b)=cosa cosb+sinasinb` so

`cosz = cos(cos^-1x)cos(cos^-1y)+sin(cos^-1x)sin(cos^-1y)`

but

`cos(cos^-1x)=x` and `sin(cos^-1x)=sqrt(1-x^2)` so

`cosz=x y + sqrt(1-x^2)sqrt(1-y^2)`

so

`cosz=1/4*2/3+sqrt(1-(1/4)^2)*sqrt(1-(2/3)^2)`

= `1/6+sqrt(15/16)*sqrt(5/9)=1/6+5/12sqrt3`

= `1/12(2+5sqrt(3))`

so `z = cos^-1(1/12(2+5sqrt(3)))`