Prove 8 ( 1 − cos 2 ⁡ a ) ( 1 − cos 2 ⁡ b ) ( 1 − cos 2 ⁡ c ) ≥ 27 cos ⁡ a cos ⁡ b cos ⁡ c

Question

Prove   8  (  1  −      cos    2    ⁡  a  )  (  1  −      cos    2    ⁡  b  )  (  1  −      cos    2    ⁡  c  )  ≥  27  cos  ⁡  a  cos  ⁡  b  cos  ⁡  c

arhaitategr · Accepted Answer

Let u, v and w be lengths-sides of the triangle such that

\cos a = \frac{v^{2} + w^{2} - u^{2}}{2 v w}

,

\cos b = \frac{u^{2} + w^{2} - v^{2}}{2 u w}

,

\cos c = \frac{u^{2} + v^{2} - w^{2}}{2 u v}

and
let

u^{2} + v^{2} - w^{2} = r

,

u^{2} + w^{2} - v^{2} = q

,

v^{2} + w^{2} - u^{2} = p

Hence, we need to prove that

8 \prod_{c y c} (1 - \frac{r^{2}}{(p + r) (q + r)}) \geq \frac{27 p q r}{(p + q) (p + r) (q + r)}

or

8 (p q + p r + q r)^{3} \geq 27 p q r (p + q) (p + r) (q + r) .

Let pq=z, pr=y and qr=x.
Hence, we need to prove that

8 (x + y + z)^{3} \geq 27 (x + y) (x + z) (y + z),

which is AM-GM:

{(\frac{\sum_{c y c} (x + y)}{3})}^{3} \geq \prod_{c y c} (x + y) .

Prove 8 ( 1 − cos 2 </msup> ⁡

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