<s>[INST] Solutions of $\sin^2\theta = \frac{x^2+y^2}{2xy} $ If $x$ and $y$ are real, then the equation
$$\sin^2\theta = \frac{x^2+y^2}{2xy}$$
has a solution:


*

*for all $x$ and $y$  

*for no $x$ and $y$  

*only when $x \neq y \neq 0$  

*only when $x = y \neq 0$ 

 [/INST] $$\begin{align}
  & {{\sin }^{2}}\theta =\frac{1}{2}\left( \frac{x}{y}+\frac{y}{x} \right)\ge 1\,\,\,\,\quad,\,\,\,\,\,\,\,\,\,\frac{x}{y}>0\,\,\,\,\,\,\Rightarrow \,\,\,\,\theta =k\pi +\frac{\pi }{2}\, \\ 
 & {{\sin }^{2}}\theta =\frac{1}{2}\left( \frac{x}{y}+\frac{y}{x} \right)\le -1\,\,\,\,\,,\,\,\,\frac{x}{y}<0\,\,\,\,\,\, \\ 
\end{align}$$ 
</s>