The period of $\sin \left(\sin x\right)+\sin \left(\cos x\right)$ is k , then $\frac{2\pi }{k}$  is

Period of $\sin \left(\cos x\right)=2\pi$

$\therefore$  Period of $\sin \left(\sin x\right)+\sin \left(\cos x\right)=\text{L}\text{.C}\text{.M}\left(2\pi ,2\pi \right)=2\pi =k$  

$\therefore \frac{2\pi }{k}=\frac{2\pi }{2\pi }=1$