How do I simplify ##sin 45 cos 15 + cos 45 sin 15 ## with the angles in degrees?

Regardless of the unit you use for the angle, the following relations hold:

##\sin(a + b)=\sin(a)\cos(b) + \cos(b)\sin(a)## ##\sin(a - b)=\sin(a)\cos(b) - \cos(b)\sin(a)## ##\cos(a + b) = \cos(a)\cos(b) - \sin(a)\sin(b)## ##\cos(a - b) = \cos(a)\cos(b) + \sin(a)\sin(b)##

(you can check them out here )

You only need to recognize the right case: ##\sin(45)\cos(15) + \cos(45)\sin(15)## is an expression of the form ##\sin(a)\cos(b) + \cos(b)\sin(a)##, which is the sine of the sum of the angles, so ##\sin(45)\cos(15) + \cos(45)\sin(15)=\sin(45+15)=\sin(60)=\sqrt(3)/2##