Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $\cos^2 x+\cos^2 y +\cos^2 z+\cos^2 t=1.$Find the minimum possible value of $\cot x+\cot y+\cot z+\cot t.$ Nice problem.

ThE-dArK-lOrD wrote: Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ https://artofproblemsolving.com/community/c6h1498328p8825595

ThE-dArK-lOrD wrote: Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ It's one of the most beautiful and intelligent problems I saw recently . I'll come up with a solution later. The answer is 2 .

I can tell $\cos(x)+\cos(y)+\cos(z)+\cos(t)\ge 1$.

ThE-dArK-lOrD wrote: Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ You can tell a lot . But the solution is this .

Attachments:

From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$

mihaig wrote: ThE-dArK-lOrD wrote: Given real numbers $x,y,z,t\in (0,\pi /2]$ such that $$\cos^2 (x)+\cos^2 (y) +\cos^2 (z) +\cos^2 (t)=1.$$What is the minimum possible value of $$\cot (x) +\cot (y) +\cot (z) +\cot (t)?$$ You can tell a lot . But the solution is this . For the last past, using GM-HM, $\sqrt{1\cdot\frac{u}{a+b}}\ge\frac{2}{1+\frac{a+b}{u}}=\frac{2u}{u+a+b}$. Adding these up, we get the result. Edit: Thinking about it, we can use this method for the whole problem. Letting $a=\cos^2t$, we need to prove that $\sum_{ }^{ }\sqrt{\frac{a}{b+c+d}}\ge2$. But by GM-HM $\sqrt{1\cdot\frac{a}{b+c+d}}\ge\frac{2}{1+\frac{b+c+d}{a}}=\frac{2a}{a+b+c+d}$. So adding these up, we are done.

Gems98 wrote: From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$ Nothing compares to this one .

Gems98 wrote: From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$ That's a marvel.

Gems98 wrote: From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$ I think, it deserves an upvote!

Gems98 wrote: From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$ When does equality occur?

SHARKYKESA wrote: Gems98 wrote: From $sin(x)cos(x) \leqslant \frac{1}{2}$ So, $cot(x) \geqslant 2 cos^2(x)$ Therefore, $cot(x)+cot(y)+cot(z)+cot(t) \geqslant 2$ When does equality occur? For example, when $x=y=\frac{\pi}{4}$ and $z=t=\frac{\pi}{2}$.