Math Marathon Warmup

It is necessary to know that $2\sin x \cos x=\sin 2x$ or in another form, $\sin x \cos x =\frac{1}{2} \sin 2x$ $10^{2^2} \sin(10^{\circ})\cos(2\cdot 10^{\circ})\sin(3\cdot 10^{\circ})\cos(4\cdot 10^{\circ})$ $= 10000 \frac{\sin(10^{\circ})\cos(10^{\circ})\cos(20^{\circ})\sin(30^{\circ})\cos(40^{\circ})}{\cos(10^{\circ})}$ $= 5000\frac{\sin(20^{\circ})\cos(20^{\circ})\sin(30^{\circ})\cos(40^{\circ})}{\cos(10^{\circ})}$ $= 2500\frac{\sin(40^{\circ})\cos(40^{\circ}\sin(30^{\circ})}{\cos(10^{\circ})}$ $= 1250\frac{\sin(80^{\circ})\sin(30^{\circ})}{\cos(10^{\circ})}$ $= 1250\frac{\cos(10^{\circ})\sin(30^{\circ})}{\cos(10^{\circ})}$ (as $\sin x=\cos(90^{\circ}-x)$ $= 1250\sin(30^{\circ})$ $= 1250 \cdot \frac{1}{2}=625 \blacksquare$