By definition, the set of functions of $x$ in real domain $\{ u_{1},...,u_{n}\}$ is linearly independent $\iff$
$\exists\ \alpha_{i}\neq 0,\ i \in\{1,...,n\}\Rightarrow \alpha_{1}u_{1}+\ldots+\alpha_{n}u_{n} \neq 0 \ \forall \ x \in \mathbb{R}$.
From the set of functions, I can write:
$1\alpha+\beta\sin^2(x)+\gamma\cos^2(x)=0 \Rightarrow (\beta - \gamma)\sin^2(x) + \alpha + \gamma=0$
Why I can't take $\,x=\arcsin\sqrt{\dfrac{-\alpha-\gamma}{\beta-\gamma}}\,$ as a counter example, showing linear dependence for the set?
The set $\left\{1,\sin^2(x),\cos^2(x)\right\}$ is linearly dependent, indeed there exist $\;\alpha=-1\neq0$, $\;\beta=\gamma=1\neq0\;$ such that $$\alpha\cdot1+\beta\sin^2(x)+\gamma\cos^2(x)=0\;,\quad\forall\,x\in\Bbb R\,.$$