Need help with inequalities

[Sentendence]

I just remembered something I proved when I was 14, but the problem is that I don't remember them exactly the way I proved them.

Could someone please check whether \(\displaystyle \begin{vmatrix}
x &y \\
1& 1
\end{vmatrix}^{2}\leq x^{4}+y^{4}\) is true for all real \(\displaystyle x\)?

I believe I did this by using the vector product, by considering \(\displaystyle \sin x\), \(\displaystyle \cos x\), \(\displaystyle \tan x\) and \(\displaystyle \cot x\).

Thank you very much!

 

[lookagain]

I just remembered something I proved when I was 14, but the problem is that I don't remember them exactly the way I proved them.

Could someone please check whether \(\displaystyle \begin{vmatrix}
x &y \\
1& 1
\end{vmatrix}^{2}\leq x^{4}+y^{4}\) is true for all real \(\displaystyle x\)?

I believe I did this by using the vector product, by considering \(\displaystyle \sin x\), \(\displaystyle \cos x\), \(\displaystyle \tan x\) and \(\displaystyle \cot x\).

Thank you very much!

So, the left-hand side is the square of a determinant of a matrix, yes? Do you know how to compute the determinant of that matrix? If so, what is the truth value of the inequality if x = 1 and y = -1?

 

[Sentendence]

So, the left-hand side is the square of a determinant of a matrix, yes? Do you know how to compute the determinant of that matrix? If so, what is the truth value of the inequality if x = 1 and y = -1?

Oh, I'm sorry, I think I proved it for positive real numbers then.

First, you need to give us the answer to this 26 terms multiplication:

(a-n)(b-n) .... (y-n)(z-n) = ?

I don't understand how that is related to my question, but that is 0.