$\text{ If the periods of the functions }\sin (ax+b)\text{ and }$$\tan (\mathrm{cx}+\mathrm{d})\text{ are respectively }\frac{1}{7}\text{ and }\frac{2}{5}\text{ then }$

$\sin \left(\right|a|+|c\left|\right)+\cos \left(\right|a|-|c\left|\right)=$

$\sin (ax+b)=\frac{2\pi }{\left|a\right|}=\frac{4}{7}\Rightarrow \left|a\right|=\frac{7\pi }{2}$

$\text{ Period of }\tan (cx+d)=\frac{\pi }{\left|c\right|}=\frac{2}{5}\Rightarrow \left|c\right|=\frac{5\pi }{2}\text{, }$

$\Rightarrow \sin \left(\right|a|+|c\left|\right)+\cos \left(\right|a|-|c\left|\right)=\sin \left(\frac{7\pi }{2}+\frac{5\pi }{2}\right)+\cos \left(\frac{7\pi }{2}-\frac{5\pi }{2}\right)$

$=\sin 6\pi +\cos \pi =0-1=-1$