Main Identities i'll be using here are,
#color(red)(cosh x =(e^x + e^-x)/2# #color(white)(rrr# and #color(white)(rrr# #color(magenta)(sinh x=(e^x-e^-x)/2#
To Prove #->##sinh(a+b)=sinhacoshb+sinhbcosha#
I'll begin with the right side,
#color(white)(rrr#
#=>color(magenta)(sinha)color(red)(coshb)+color(magenta)(sinhb)color(red)(cosha#
#=>(color(magenta)((e^a-e^-a)/2) xx color(red)((e^b+e^-b)/2))+(color(magenta)((e^b-e^-b)/2) xx color(red)((e^a+e^-a)/2))#
#color(white)(rrr#
#=>((e^a-e^-a)(e^b+e^-b))/4+((e^b-e^-b)(e^a+e^-a))/4#
#color(white)(rrr#
#=>1/4[e^(a+b)+e^(a-b)-e^(b-a)-e^(-(a+b))+ e^(a+b)-e^(a-b)+e^(b-a)-e^(-(a+b))]#
#color(white)(rrr#
#=>1/4[e^(a+b)+cancel(e^(a-b))-cancel(e^(b-a))-e^(-(a+b))+ e^(a+b)cancel(-e^(a-b))cancel(+e^(b-a))-e^(-(a+b))]#
#color(white)(rrr#
#=>1/4[2e^(A+B)-2e^(-(A+B))]#
#color(white)(rrr#
#=>1/cancel4^2[cancel2(e^(A+B)-e^(-(A+B))]#
#color(white)(rrr#
#=>(e^(A+B)-e^(-(A+B)))/2#
#color(white)(rrr#
#=>sinh(A+B)#
Hence Proved :)
