Finding the minimum value of $\cot^2A + \cot^2B+ \cot^2C$ where $A$, $B$ and $C$ are angles of a triangle. The question is:

If $A+B+C= \pi$, where $A>0$, $B>0$, $C>0$, then find the minimum value of $$\cot^2A+\cot^2B +\cot^2C.$$

My solution: 
$(\cot A + \cot B + \cot C)^2\ge0$   //  square of a real number
$\implies \cot^2A +\cot^2B + \cot^2 +2 \ge0   $  //Conditional identity used: $\cot A \cot B + \cot B \cot C + \cot A \cot C =1$
$\implies \cot^2A +\cot^2B + \cot^2 C \ge -2$
Thus according to me the answer should be $-2$. However, the answer key states that the answer is $1$. Where have I gone wrong? 

Since $\cot^2$ is a convex function, we can use Jensen.
$$\sum_{cyc}\cot^2\alpha\geq3\cot^2\frac{\alpha+\beta+\gamma}{3}=1.$$
The equality occurs for $\alpha=\beta=\gamma=\frac{\pi}{3},$ which says that $1$ is a minimal value.
Actually, $(\cot^2x)''=\frac{2(2+\cos2x)}{\sin^4x}>0$.
Also, we can use the following way.
Let $a$, $b$ and $c$ be sides-lengths and $S$ be an area of the triangle.
Thus, we need to prove that
$$\sum_{cyc}\frac{\cos^2\alpha}{\sin^2\alpha}\geq1$$ or
$$\sum_{cyc}\frac{\frac{(b^2+c^2-a^2)^2}{4b^2c^2}}{\frac{4S^2}{b^2c^2}}\geq1$$ or 
$$\sum_{cyc}(b^2+c^2-a^2)^2\geq16S^2$$ or
$$\sum_{cyc}(b^2+c^2-a^2)^2\geq\sum_{cyc}(2a^2b^2-a^4)$$ or
$$\sum_{cyc}(a^4-a^2b^2)\geq0$$ or
$$\sum_{cyc}(a^2-b^2)^2\geq0.$$
Done!
Also we can make the following. It's what you wish. I think.
$$\sum_{cyc}\cot^2\alpha=1+\sum_{cyc}(\cot^2\alpha-\cot\alpha\cot\beta)=1+\frac{1}{2}\sum_{cyc}(\cot\alpha-\cot\beta)^2\geq1$$
It's
$$\cot^2\alpha+\cot^2\beta+\cos^2\gamma=$$
$$=1+\cot^2\alpha+\cot^2\beta+\cos^2\gamma-\cot\alpha\cot\beta-\cot\alpha\cot\gamma-\cot\beta\cot\gamma=$$
$$=1+$$
$$+\frac{1}{2}\left(\cot^2\alpha-2\cot\alpha\cot\beta+\cot^2\beta+\cot^2\alpha-2\cot\alpha\cot\gamma+\cos^2\gamma+\cot^2\beta-2\cot\beta\cot\gamma+\cot^2\gamma\right)=$$
$$=1+\frac{1}{2}\left((\cot\alpha-\cot\beta)^2+(\cot\alpha-\cot\gamma)^2+(\cot\beta-\cot\gamma)^2\right)\geq1.$$
