If the periods of the functions   $\sin (ax+b)$ and   $\tan (ax+b)$ are respectively $\frac{4}{7}\text{ and }\frac{2}{5}$ then$\sin \left(\right|a|+|c\left|\right)+\cos \left(\right|a|-|c\left|\right)=$

$$\begin{gathered} Period of \sin (ax+b)=\frac{2\mathrm{\pi }}{\left|a\right|}=\frac{4}{7}\Rightarrow \left|a\right|=\frac{7\mathrm{\pi }}{2}, \\ Period of \tan \left(cx+d\right)=\frac{\mathrm{\pi }}{\left|c\right|}=\frac{2}{5}\Rightarrow \left|c\right|=\frac{5\mathrm{\pi }}{2}, \\ \Rightarrow \sin \left(\right|a|+|c\left|\right)+\cos \left(\right|a|-|c\left|\right)=\sin \left(\frac{7\mathrm{\pi }}{2}+\frac{5\mathrm{\pi }}{2}\right)+\cos \left(\frac{7\mathrm{\pi }}{2}-\frac{5\mathrm{\pi }}{2}\right) \\ =\sin 6\mathrm{\pi }+\mathrm{cos\pi }=0-1=-1 \end{gathered}$$