$\text{ If }\frac{\cos \left({\theta }_1-{\theta }_2\right)}{\cos \left({\theta }_1+{\theta }_2\right)}+\frac{\cos \left({\theta }_3+{\theta }_4\right)}{\cos \left({\theta }_3-{\theta }_4\right)}=\mathbf{0}\text{ then }\tan {\theta }_1\cdot \tan {\theta }_2\cdot \tan {\theta }_3\cdot \tan {\theta }_4=$

$$\begin{gathered} \frac{\cos \left({\theta }_1-{\theta }_2\right)}{\cos \left({\theta }_1+{\theta }_2\right)}+\frac{\cos \left({\theta }_3+{\theta }_4\right)}{\cos \left({\theta }_3-{\theta }_4\right)}=\mathbf{0} \\ \mathbf{\Rightarrow }\frac{\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}}{\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}}\mathbf{=}\frac{\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}}{\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}} \\ \mathbf{using}\mathbf{ }\mathbf{componendo}\mathbf{ }\mathbf{and}\mathbf{ }\mathbf{dividendo} \\ \mathbf{\Rightarrow }\frac{\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}\mathbf{+}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}}{\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{2}}\mathbf{)}}\mathbf{=}\frac{\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}\mathbf{+}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}}{\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{+}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}\mathbf{-}\mathbf{cos}\mathbf{}\mathbf{(}{\mathbf{\theta }}_{\mathbf{3}}\mathbf{-}{\mathbf{\theta }}_{\mathbf{4}}\mathbf{)}} \\ \mathbf{\Rightarrow }\frac{\mathbf{2}\mathbf{cos}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{cos}{\mathbf{\theta }}_{\mathbf{2}}}{\mathbf{2}\mathbf{sin}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{sin}{\mathbf{\theta }}_{\mathbf{2}}}\mathbf{=}\frac{\mathbf{2}{\mathbf{sin\theta }}_{\mathbf{3}}{\mathbf{sin\theta }}_{\mathbf{4}}}{\mathbf{-}\mathbf{2}{\mathbf{cos\theta }}_{\mathbf{3}}{\mathbf{cos\theta }}_{\mathbf{4}}} \\ \mathbf{\Rightarrow }\frac{\mathbf{1}}{\mathbf{tan}{\mathbf{\theta }}_{\mathbf{1}}\mathbf{tan}{\mathbf{\theta }}_{\mathbf{2}}}\mathbf{=}\mathbf{-}{\mathbf{tan\theta }}_{\mathbf{3}}{\mathbf{tan\theta }}_{\mathbf{4}} \\ \mathbf{\therefore }{\mathbf{tan\theta }}_{\mathbf{1}}{\mathbf{tan\theta }}_{\mathbf{2}}{\mathbf{tan\theta }}_{\mathbf{3}}{\mathbf{tan\theta }}_{\mathbf{4}}\mathbf{=}\mathbf{-}\mathbf{1} \end{gathered}$$