(sin1)(sin2)(sin3)...

[Haile]

Si dimostri che

$ $\prod_{j=1°}^{90°} \sin(j) = \frac{6\sqrt{10}}{4^{45}}$ $

Idee?

[pak-man]

$ $\sin\alpha\sin{\left(\frac{\pi}{2}-\alpha\right)}=\sin\alpha\cos\alpha=\frac{1}{2}\sin{2\alpha} $
$ $\prod_{j=1^{\circ}}^{90^{\circ}}\sin{j}=\frac{\sqrt{2}}{2^{45}}\prod_{i=1^{\circ}}^{44^{\circ}}\sin{2i} $

Idee?

[Haile]

$ $\sin\alpha\sin{\left(\frac{\pi}{2}-\alpha\right)}=\sin\alpha\cos\alpha=\frac{1}{2}\sin{2\alpha} $
$ $\prod_{j=1^{\circ}}^{90^{\circ}}\sin{j}=\frac{\sqrt{2}}{2^{45}}\prod_{i=1^{\circ}}^{44^{\circ}}\sin{2i} $

Idee?

$ $\sin(2°) \sin(4°) \dots \sin(88°)$ $

$ $(\sin(2°) \cos(2°))(\sin(4°) \cos(4°)) \dots (\sin(44°) \cos(46°))$ $

$ $\prod_{j=1^{\circ}}^{90^{\circ}}\sin{j} = \frac{\sqrt{2}}{2^{45}}\prod_{i=1^{\circ}}^{44^{\circ}}\sin{2i} = \frac{\sqrt2}{2^{45}} \cdot \frac{1}{2^{22}} \prod_{k=1}^{22°} \sin(4k)$ $

Idee?

[Boll]

HINT: Radici 180-esime dell'unità