can someone please help me with the proof of minimum value of a^2tan^2theta + b^2cot^2theta = 2ab

[alpine pike]

Minimum value of what? theta? 🤔

[grizzled flame]

no the proof of the whole thing

the minimum value of a^2tan^2theta + b^2cot^2theta is 2ab

i need the proof of this

[alpine pike]

notice that you can get a binomial squared

and anything squared is always >= 0

[coral birch]

Let $x = \theta$ We want to proof $$a^2 \cdot tan^2(x) + b^2 \cdot cot^2(x) \ge 2ab $$
Notice that $tan(x) \cdot cot(x) =1$ we can rewrite

$$a^2 \cdot tan^2(x) + b^2 \cdot cot^2(x) \ge 2ab\cdot tan(x) \cdot cot(x) $$
$$\iff a^2 \cdot tan^2(x) - 2ab\cdot tan(x) \cdot cot(x) + b^2 \cdot cot^2(x) \ge 0$$

$$\iff \left(a\cdot tan(x) - b\cdot cot(x)\right)^2 \ge 0$$
But a square is always positiv hence we proved the inequality

[sullen sedge]

Homelama