You have sin B=8/17 and cosA=12/13.
From this information, compute cosB and sin A
sinA=sqrt(1-cos^2A)=5/13
cos B=sqrt(1-sin^2B)=15/17
So, tanA=sinA/cosA=5/12 and
tanB=sinB/cosB=8/15
color(blue) sin(A+B) color(blue)= color(blue)(sinAcosB+cosAsinB
color(blue)(=5/13*15/17+12/13*8/17= 171/221
color(green)(cos(A+B)=sinAsinB-cosAcosB
color(green)(=12/13*15/17-5/13*8/17=140/221
color(brown)(tan(A+B)=(tanA+tanB)/(1-tanAtanB)
color(brown)(=(5/12+8/15)/(1-5/12*8/15)=171/180*9/7=171/140