$\begin{aligned} & \text { Given } f(x)=\cos (\sin \mathrm{h}(\log x)+\cos \mathrm{h}(\log x)) \\ & f(x)=\cos \left(\frac{e^{\log x}-e^{-\log x}+e^{\log x}+e^{-\log x}}{2}\right) \\ & =\cos \left(\frac{\left.x-\frac{1}{x}+x+\frac{1}{x}\right)}{2}\right)=\cos x \\ & f(x)=\cos x \\ & \text { Differentiate w.r.t. } x \\ & f^{\prime}(x)=-\sin x \\ & f^{\prime}(x)=\sin x=0 ; x=\pi \\ & f^{\prime \prime}(x)=-\cos x \\ & f^{\prime}(\pi)=1>0(\text { Minimum value }) \\ & \text { Minimum value of } f(x)=\cos \pi=-1 \\ & \text { Then, } k=-1 \\ & \text { Now, } \cos \mathrm{h}(k+1)=\cos h(-1+1)=\cos 0=1\end{aligned}$