Without using tables, prove that $$\left( {\sin \,{{12}^ \circ }} \right)\left( {\sin \,{{48}^ \circ }} \right)\left( {\sin \,{{54}^ \circ }} \right) = {1 \over 8}.$$

The inequality |z-4| < |z-2| represents the region given by

Show that the equation $${e^{\sin x}} - {e^{ - \sin x}} - 4 = 0$$ has no real solution.

The largest interval for which $${x^{12}} - {x^9} + {x^4} - x + 1 > 0$$ is