Minimizing $\tan^2 x+\frac{\tan^2 y}{4}+\frac{\tan^2 z}{9}$ 
Given that $\tan x+2\tan y+3\tan z=40 , \ \ \ x,y,z \in \left(\dfrac{\pi}{2},\dfrac{3\pi}{2}\right),$ 
We need to find the minimum value of $ \tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9}$

One way would be to consider the vectors 
$$\begin{align}
&\vec{v_1}=\tan x \ \hat{i}+\dfrac{\tan y}{2} \ \hat{j}+\dfrac{\tan z}{3} \ \hat{k} \ \ \ \ \ \ \ \ \text{and}\\ &\vec{v_2}=\hat{i}+4 \ \hat{j}+9 \ \hat{k}
\end{align}$$
And then apply to them the inequality $\left(\vec{v_1} \cdot \vec{v_2}\right)^2 \leq \left|\vec{v_1}\right|^2 \left|\vec{v_2}\right|^2$.
Using this we get $\tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9} \geq \dfrac{1600}{98}$
Is this method fine?
What are other methods to solve this problem, which do not involve vectors or complex numbers? Can this result be obtained by using elementary calculus?
Thank you.

Define $f:\left(\frac{\pi}{2}, \frac{3\pi}{2}\right)^3\to \Bbb R$ putting $$f(x,y,z) =  \tan^2 x+\dfrac{\tan^2 y}{4}+\dfrac{\tan^2 z}{9} $$
You want to optimize $f$ subject to $$g(x,y,z) = \tan x+2\tan y+3\tan z=40$$
Call $u = \tan x$, $v = \tan y$ and $w = \tan z$. Since $x,y,z \in \left(\frac{\pi}{2},\frac{3\pi}{2}\right)$, we have that $u,v,w \in \Bbb R$.
It seems easier to me work with: $$\tilde{f}(u,v,w) = u^2 + \frac{v^2}{4} + \frac{w^2}{9}$$ and the restriction: $$\tilde{g}(u,v,w) = u + 2v + 3w = 40$$
Now use Lagrange Multipliers and solve: $$\begin{cases} \nabla \tilde{f}(u,v,w) = \lambda \nabla \tilde{g}(u,v,w) \\ \tilde{g}(u,v,w) = 40 \end{cases}.$$
Your method seems fine to me, though.
