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#