Review question

# Can we prove $\sin a \sin b \leq \sin^2\frac{1}{2}(a+b)$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8525

## Question

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.