Let sin^(-1)(4/5)=x then
rarrsinx=4/5
rarrtanx=1/cotx=1/(sqrt(csc^2x-1))=1/(sqrt((1/sinx)^2-1))=1/(sqrt((1/(4/5))^2-1))=4/3
rarrx=tan^(-1)(4/3)=sin^(-1)=(4/5)
Now,
rarrcos(sin^(-1)(4/5)+tan^(-1)(5/12))
=cos(tan^(-1)(4/3)+tan^(-1)(5/12))
=cos(tan^(-1)((4/3+5/12)/(1-(4/3)*(5/12))))
=cos(tan^(-1)((63/36)/(16/36)))
=cos(tan^(-1)(63/16))
Let tan^(-1)(63/16)=A then
rarrtanA=63/16
rarrcosA=1/secA=1/sqrt(1+tan^2A)=1/sqrt(1+(63/16)^2)=16/65
rarrA=cos^(-1)(16/65)=tan^(-1)(63/16)
rarrcos(sin^(-1)(4/5)+tan^(-1)(5/12))=cos(tan^(-1)(63/16))=cos(cos^(-1)(16/65))=16/65