When we divide complex numbers we divide their magnitudes and subtract their angles, so this isn't all that difficult:
[tex]\dfrac{10(\cos 45^\circ + i \sin 45^\circ)}{5 (\cos 15^\circ + i \sin 15^\circ)} = \dfrac{10}{5} (\cos(45^\circ - 15^\circ) + i \sin(45^\circ - 15^\circ)) [/tex]
[tex]= 2(\cos 30^\circ + i \sin 30^\circ)[/tex]
Answer: [tex] 2(\cos 30^\circ + i \sin 30^\circ)[/tex]
That's probably what's called simplified polar form until you get to Euler's Formula. Since it's 30 degrees we can get a nice rectangular form:
[tex]2(\cos 30^\circ + i \sin 30^\circ) = 2\left( \dfrac{\sqrt{3}}{2} + i \dfrac 1 2 \right) = \sqrt{3} + i [/tex]
