Find the minimum value of $\tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$

$x$, $y$ and $z$ are three acute angles from a triangle.

Find the minimum value of $\tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$.

The trick for solving this problem fast and effective depends on if you could use the implicit relation between $x,\,y$ and $z$ when they are the angles from a triangle:

$\tan x+\tan y+\tan z=\tan x \tan y \tan z$

By using AM-GM inequality, we know $\tan x+\tan y+\tan z\ge 3\sqrt[3]{\tan x \tan y \tan z}$. Replace $\tan x+\tan y+\tan z$ by $\tan x \tan y \tan z$, we get:

$\tan x \tan y \tan z\ge 3\sqrt[3]{\tan x \tan y \tan z}$

$\tan x \tan y \tan z\ge 3\sqrt{3}$

$\tan x \tan y \tan z\ge (3)^{\frac{3}{2}}$

Going back to the objective function, we need to minimize

$\tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$

We attempt at doing that by replacing

$\tan y \tan z$ by $\dfrac{\tan x+\tan y+\tan z}{\tan x}$

$\tan x \tan z$ by $\dfrac{\tan x+\tan y+\tan z}{\tan y}$

$\tan x \tan y$ by $\dfrac{\tan x+\tan y+\tan z}{\tan z}$

$\displaystyle \tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$

$\displaystyle =\tan^7 x\left(\dfrac{\tan x+\tan y+\tan z}{\tan x}-1\right)+\tan^7 y\left(\dfrac{\tan x+\tan y+\tan z}{\tan y}-1\right)+\tan^7 z \left(\dfrac{\tan x+\tan y+\tan z}{\tan z}-1\right)$

$\displaystyle =\tan^6 x(\tan x+\tan y+\tan z)-\tan^7 x+\tan^6 y(\tan x+\tan y+\tan z)-\tan^7 y+\tan^6 z(\tan x+\tan y+\tan z)-\tan^7 z$

$\displaystyle =(\tan^6 x+\tan^6 y+\tan^6 z)(\tan x+\tan y+\tan z)-(\tan^7 x+\tan^7 y+\tan^7 z)$

$\displaystyle =(\tan^7 x+\tan^7 y+\tan^7 z+\tan^6x \tan y+\tan^6x \tan z+\tan^6y \tan x+\tan^6y \tan z+\tan^6z \tan x+\tan^6z \tan y)-(\tan^7 x+\tan^7 y+\tan^7 z)$

$\displaystyle =\tan^6x \tan y+\tan^6x \tan z+\tan^6y \tan x+\tan^6y \tan z+\tan^6z \tan x+\tan^6z \tan y$

Now, apply the AM-GM inequality we obtain:

$\displaystyle \tan^7 x(\tan y \tan z-1)+\tan^7 y(\tan x\tan z-1)+\tan^7 z(\tan x \tan y-1)$

$\displaystyle =\tan^6x \tan y+\tan^6x \tan z+\tan^6y \tan x+\tan^6y \tan z+\tan^6z \tan x+\tan^6z \tan y$

$\displaystyle \ge 6\sqrt[6]{\tan^{14} x\tan^{14} y\tan^{14} z}$

$\displaystyle \ge 6(\tan^{14} x\tan^{14} y\tan^{14} z)^{\frac{14}{6}}$

$\displaystyle \ge 6(\tan^{14} x\tan^{14} y\tan^{14} z)^{2\frac{1}{3}}$

$\displaystyle \ge 6((3)^{\frac{3}{2}})^{2\frac{1}{3}}$

$\displaystyle \ge 6((3)^{\frac{7}{2}})$

$\displaystyle \ge 6(3^3\sqrt{3})$

$\displaystyle \ge 162\sqrt{3})$

Equality attains when $\tan x=\tan y=\tan z$, i.e. $\tan x+\tan x+\tan x=\tan x\tan x\tan x$, $3\tan x=\tan^3 x$, $\tan x=+\sqrt{3}=\tan y=\tan z$.