Prove that, for all values of \(a\) and \(b\), \[\sin a \sin b \le \sin^2\frac{1}{2}(a+b).\]
Show further that, if \(a\), \(b\), \(c\) and \(d\) all lie between \(0\) and \(\pi\), then \[\sin a \sin b \sin c \sin d \leq \left( \sin\frac{1}{4}(a+b+c+d)\right)^4;\] and, by writing \(d=\frac{1}{3}(a+b+c)\), deduce that \[\sin a \sin b \sin c \le \left( \sin\frac{1}{3}(a+b+c)\right)^3.\]
Show precisely where the restrictions on \(a\), \(b\), \(c\) and \(d\) are used in the proof of the last two inequalities.
