$$\begin{gathered} Given sec{h}^2\left(\tan h^{-1}\right(\frac{1}{2}\left)\right)+cosec{h}^2\left(cot{h}^{-1}\right(3\left)\right)=\frac{35}{k} \\ \Rightarrow 1-{\mathrm{Tanh}}^2 \left({\mathrm{Tanh}}^{-1} \frac{1}{2}\right)+{\coth }^2 \left({\coth }^{-1} 3\right)-1=\frac{35}{k}(\because sec{h}^2\theta +\tan h^2{\theta }^{ }=1 and cot{h}^2\theta -\cos ec{h}^2\theta =1) \\ \Rightarrow 1-{\left(\frac{1}{2}\right)}^2+{\left(3\right)}^2-1=\frac{35}{k} \\ \Rightarrow 9-\frac{1}{4}=\frac{35}{k} \\ \Rightarrow \frac{35}{4}=\frac{35}{k} \\ \Rightarrow k=4 \end{gathered}$$