Minimum how?

Geometry Level 3

The minimum value of $3 \tan^2 (A) + 12 \cot^2 (A)$ is?

2 solutions

Ap V
Jul 23, 2015

Its easy by am- gm method.

Jason Zou
Jul 6, 2015

Let $x=\tan(A)$ .

We see the expression becomes $3x^2+\frac{12}{x^2}$

Factor out the $3$ to get $3(x^2+\frac{4}{x^2})$

That expression is clearly minimum when $x^2=2$ because $x^2$ must be positive.

Note that it is possible for $\tan(A)=\sqrt{2}$

Plugging $x^2=2$ in, we get $3(2+\frac{4}{2})=\boxed{12}$