On which open intervals are the following functions linearly dependent: $f(x) = \operatorname{cosh}^2x$, $g(x) = 2$, $h(x) = \operatorname{sinh}^2x?$
I tried wronskian (to find out whether independent) and it turned out to be $0$ so it doesn't tell me anything (assuming I calculated correctly).
I tried solving straight from definition - that is checking if for some interval $I \subset \mathbb{R}$ there are real ${\alpha}, {\beta}, {\gamma}$, not all of them zeros, such that ${\alpha}\operatorname{cosh}^2x + 2{\beta} + {\gamma}\operatorname{sinh}^2x = 0$ for all $x \in I$. I could not make it work. Then I checked for the simpler linear dependence ${\alpha}\operatorname{cosh}^2x + {\beta}\operatorname{sinh}^2x = 0$ and wound up with the equation $e^{2x} + e^{-2x} = {\frac{2(\beta-\alpha)}{\beta+\alpha}}$.
Any advice on what to do?
Hint:
$$ \cosh^2 x - \sinh^2 x = 1 $$
For every $x$. The wronskian is a nuclear option.