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

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Answer 1 · 2017-02-07T11:00:46

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

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)))#