<s>[INST] Trigonometry Inequality This is the first time I'm posting here. If you can also tell me how to format this like a pro, I'll be very grateful.
1st question:
Prove the following inequality:
$$0^{\circ} < a, b, c < 180^{\circ}$$
$$\sin a \times \sin b \times \sin c \le \sin\left(\frac{a+b}{2}\right) \times \sin\left(\frac{a+c}{2}\right) \times \sin\left(\frac{a+b}{2}\right)$$
2nd question:
Prove the following inequality:
$$a + b + c = 90^{\circ}$$
$$\sin a \times \sin b \times \sin c  \le \frac{1}{8}$$
 [/INST] First inequality
By Jensen we have
$$\sin\left(\frac{a+b}{2}\right) \geq \frac{\sin(a)+\sin(b)}{2}$$
By AM-GM we get
$$\frac{\sin(a)+\sin(b)}{2} \geq \sqrt{ \sin(a) \sin(b)}$$
Combining we get
$$\sqrt{\sin(a) \sin(b) } \leq \sin\left(\frac{a+b}{2}\right)$$
Similarly you get
$$\sqrt{\sin(a) \sin(c) } \leq \sin\left(\frac{a+c}{2}\right)$$
$$\sqrt{\sin(b) \sin(c) } \leq \sin\left(\frac{b+c}{2}\right)$$
Multiplying them you get the desired inequality.
Second Inequality:
By AM-GM:
$$\sin a \times \sin b \times \sin c  \le  \left( \frac{\sin(a)+\sin(b)+\sin(c)}{3} \right)^3$$
Now, by Jensen:
$$\frac{\sin(a)+\sin(b)+\sin(c)}{3} \leq \sin(\frac{a+b+c}{3})=\frac{1}{2}$$
Combining the two Yields the desired result.
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