This problem can be solved by repeatedly applying the triple angle formulas %%HREF%%3852%%: \begin{align} & \sin 1^\circ \sin 2^\circ \cdots \sin 89^\circ\\ =\quad&(\sin 1^\circ\sin 59^\circ\sin 61^\circ)(\sin 2^\circ\sin 58^\circ \sin 62^\circ)\cdots\\ &(\sin 29^\circ\sin 31^\circ 89^\circ)\sin 30^\circ \sin 60^\circ \\ =\quad&\big(\frac{1}{4}\big)^{29}\sin 3^\circ \sin 6^\circ \cdots \sin 87^\circ \cdot\frac{1}{2}\cdot\frac{\sqrt{3}}{2}\\ =\quad&\big(\frac{1}{4}\big)^{30}\cdot\sqrt{3}\cdot(\sin 3^\circ \sin 57^\circ \sin 63^\circ)(\sin 6^\circ \sin 54^\circ \sin 66^\circ)\cdots\\ &(\sin 27^\circ \sin 33^\circ \sin 87^\circ)\sin 30^\circ \sin 60^\circ\\ =\quad&\big(\frac{1}{4}\big)^{40}\cdot 3\cdot \sin 9^\circ \sin 18^\circ \cdots \sin 81^\circ\\ =\quad&\big(\frac{1}{4}\big)^{40}\cdot 3\cdot (\sin 9^\circ\sin 81^\circ)(\sin 18^\circ\sin 72^\circ)(\sin 27^\circ\sin 63^\circ)\\ &(\sin 36^\circ\sin 54^\circ)\sin 45^\circ\\ =\quad&\big(\frac{1}{4}\big)^{40}\cdot 3\cdot\frac{\sqrt{2}}{2}\cdot(\sin 9^\circ\cos 9^\circ)(\sin 18^\circ\cos 18^\circ)(\sin 27^\circ\cos 27^\circ)\\ &(\sin 36^\circ\cos 36^\circ)\\ =\quad&\big(\frac{1}{4}\big)^{40}\cdot 3\cdot \frac{\sqrt{2}}{2}\cdot\big(\frac{1}{2}\sin 18^\circ\big)\big(\frac{1}{2}\sin 36^\circ\big)\big(\frac{1}{2}\sin 54^\circ\big)\big(\frac{1}{2}\sin 72^\circ\big)\\ =\quad&\big(\frac{1}{2}\big)^{85}\cdot 3\cdot\sqrt{2}\cdot\sin 18^\circ\sin 36^\circ\sin 54^\circ\sin 72^\circ\\ =\quad&\big(\frac{1}{2}\big)^{85}\cdot 3\cdot\sqrt{2}\cdot\sin 18^\circ\sin 36^\circ\cos 36^\circ\cos 18^\circ\\ =\quad&\big(\frac{1}{2}\big)^{85}\cdot 3\cdot\sqrt{2}\cdot(\sin 18^\circ\cos 18^\circ)(\sin 36^\circ\cos 36^\circ)\\ =\quad&\big(\frac{1}{2}\big)^{86}\cdot 3\cdot\sqrt{2}(\sin 18^\circ\cos 18^\circ)\sin 72^\circ\\ =\quad&\big(\frac{1}{2}\big)^{86}\cdot 3\cdot\sqrt{2}(\sin 18^\circ\cos 18^\circ)\cos 18^\circ\\ =\quad&\big(\frac{1}{2}\big)^{86}\cdot 3\cdot\sqrt{2}\sin 18^\circ\cos^2 18^\circ\\ =\quad&\big(\frac{1}{2}\big)^{86}\cdot 3\cdot\sqrt{2}\cdot\sin 18^\circ(1-\sin^2 18^\circ)\\ =\quad&\big(\frac{1}{2}\big)^{84}\cdot 3\cdot\sqrt{2}\cdot\frac{\sqrt{5}-1}{4}\cdot\big(1-\big(\frac{\sqrt{5}-1}{4}\big)^2\big)\\ =\quad&\boxed{\frac{3}{2^{89}}\cdot\sqrt{10}} \end{align}