Let \([t]\) denote the greatest integer less than or equal to \(t\). If the function \(f\left(x\right)=\left{\begin{matr…

Let \([t]\) denote the greatest integer less than or equal to \(t\). If the function

\(f\left(x\right)=\left{\begin{matrix}{b}^{2}\text{sin}\left(\frac{\pi }{2}\left[\frac{\pi }{2}\left(\text{cos}x+\text{sin}x\right)\text{cos}x\right]\right),\text{   }x<0 \\ \text{                           }\frac{\sin x−\frac{1}{2}\sin 2x}{{x}^{3}}\text{            },\text{      }x>0 \\ \text{                                        }a\text{                             },\text{      }x=0\end{matrix}\right.\)

is continuous at \(x = 0\), then \(a^2+b^2\) is equal to

[JEE Main 2026, 24 Jan (Shift 2)]