Do -1 < sin(x) < 1 ∀ x ∈ R
⇒ \(\sin^2\left(x\right)\in\left[0;1\right]\) ∀ x ∈ R
Để phương trình \(\sin^2\left(x\right)=m^2-1\) có nghiệm thì
\(0\le m^2-1\le1\\ \Leftrightarrow1\le m^2\le2\\ \Leftrightarrow\left\{{}\begin{matrix}\left[{}\begin{matrix}x\ge1\\x\le-1\end{matrix}\right.\\-\sqrt{2}\le x\le\sqrt{2}\end{matrix}\right.\\ \Leftrightarrow\left[{}\begin{matrix}1\le x\le\sqrt{2}\\-\sqrt{2}\le x\le-1\end{matrix}\right.\)