First you want to let #alpha=arcsin(-5/13)# and #beta=arccos(12/13)#
So now we are looking for #color(red)cos(alpha+beta)!#
#=>sin(alpha)=-5/13" "# and #" "cos(beta)=12/13#
Recall : #cos^2(alpha)=1-sin^2(alpha)=>cos(alpha)=sqrt(1-sin^2(alpha))#
#=>cos(alpha)=sqrt(1-(-5/13)^2)=sqrt((169-25)/169)=sqrt(144/169)=12/13#
Similarly, #cos(beta)=12/13#
#=>sin(beta)=sqrt(1-cos^2(beta))=sqrt(1-(12/13)^2)=sqrt((169-144)/169)=sqrt(25/169)=5/13#
#=>cos(alpha+beta)=cos(alpha)cos(beta)-sin(alpha)sin(beta)#
Then substitue all the values obtained ealier.
#=>cos(alpha+beta)=12/13*12/13-(-5/13)*5/13=144/169+25/169=169/169=color(blue)1#
