##sqrt2/2+1/2##

Recall the unit circle:

From the unit circle we can see that at ##45^o## the position of the angle on the unit circle is at ##(sqrt2/2sqrt2/2)##

##(sqrt2/2sqrt2/2)=(costhetasintheta)##

So for ##45^o## ##sin(45^o)=sqrt2/2##

Now from the unit circle we can see that at ##60^o## the position of the angle on the unit circle is at ##(1/2sqrt3/2)##

##(1/2sqrt3/2)=(costhetasintheta)##

So for ##60^o## ##cos(60^o)=1/2##

So

##sin(45^o)+cos(60^o)=sqrt2/2+1/2##