69.
(sin(x)+-sin(y})/(cos(x)+cos(y))=tan((x+-y)/2)
separate in two:
(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)
and
(sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)
prove one:
(sin(x)+sin(y))/(cos(x)+cos(y))=tan((x+y)/2)
manipulate left side
(sin(x)+sin(y))/(cos(x)+cos(y)).....rewrite using trigonometric identities
sin(x)+sin(y)=2sin((x+y)/2)cos((x-y)/2)
cos(x)+cos(y)=2cos((x+y)/2)cos((x−y)/2)
then you have
(sin(x)+sin(y))/(cos(x)+cos(y))
=(2sin((x+y)/2)cos((x-y)/2))/2cos((x+y)/2)cos((x−y)/2)........simplify
=sin((x+y)/2))/cos((x+y)/2)...........use identity tan(x)=sin(x)/cos(x)
=tan((x+y)/2))
same is for (sin(x)-sin(y))/(cos(x)+cos(y))=tan((x-y)/2)
