If $A,B>0$ and $A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$. If $A,B>0$ and $\displaystyle A+B = \frac{\pi}{3},$ Then find Maximum value of $\tan A\cdot \tan B$.
$\bf{My\; Try::}$ Given $$\displaystyle A+ B = \frac{\pi}{3}$$ and $A,B>0$.
So we can say $$\displaystyle 0< A,B<\frac{\pi}{3}$$. Now taking $\tan $ on both side, we get
$$\displaystyle \tan(A+B) = \tan \left(\frac{\pi}{3}\right).$$ So $\displaystyle \frac{\tan A+\tan B}{1-\tan A\tan B} = \sqrt{3}$.
Now Let $\displaystyle \tan A\cdot \tan B=y\;,$ Then $\displaystyle \tan B = \frac{y}{\tan A}.$
So $$\displaystyle \frac{\tan A+\frac{y}{\tan A}}{1-y}=\sqrt{3}\Rightarrow \tan^2 A+y=\sqrt{3}\tan A-y\sqrt{3}\tan A$$
So equation $$\tan^2 A+\sqrt{3}\left(y-1\right)\tan A+y=0$$
Now for real values of $y\;,$ Given equation has real roots. So its $\bf{Discrimnant>0}$
So $$\displaystyle 3\left(y-1\right)^2-4y\geq 0\Rightarrow 3y^2+3-6y-4y\geq 0$$
So we get $$3y^2-10y+3\geq 0\Rightarrow \displaystyle 3y^2-9y-y+3\geq 0$$
So we get $$\displaystyle y\leq \frac{1}{3}\cup y\geq 3$$, But above we get $\displaystyle 0<A,B<\frac{\pi}{3}$
So We Get $$\bf{\displaystyle y_{Max.} = \left(\tan A \cdot \tan B\right)_{Max} = \frac{1}{3}}.$$
My Question is can we solve above question using $\bf{A.M\geq G.M}$ Inequality or Power Mean equality.
Thanks 

$$
\tan A\tan B=\tan A\tan\big(\frac{\pi}{3}-A\big)=\tan A.\frac{\sqrt{3}-\tan A}{1+\sqrt{3}\tan A}=1-1+\frac{\sqrt{3}\tan A-\tan^2A}{1+\sqrt{3}\tan A}\\
=1+\frac{-1-\sqrt{3}\tan A+\sqrt{3}\tan A-\tan^2A}{1+\sqrt{3}\tan A}=1-\frac{1+\tan^2A}{1+\sqrt{3}\tan A}=1-\frac{\sec^2A}{1+\sqrt{3}\tan A}\\
=1-\frac{2}{2\cos^2A+\sqrt{3}.2\sin A\cos A}=1-\frac{2}{1+2\big[\frac{1}{2}\cos 2A+\frac{\sqrt{3}}{2}\sin 2A\big]}\\
=1-\frac{2}{1+2\sin\big(\frac{\pi}{6}+2A\big)}
$$
$\tan A\tan B$ is maximum $\implies \frac{2}{1+2\sin\big(\frac{\pi}{6}+2A\big)}$ is minimum $\implies \sin\big(\frac{\pi}{6}+2A\big)$ is maximum$\implies \sin\big(\frac{\pi}{6}+2A\big)=1$
$$
(\tan A\tan B)_\text{max}=1-\frac{2}{1+2.1}=1-\frac{2}{3}=\frac{1}{3}
$$
