If $\mathrm{\alpha },\mathrm{\beta },\mathrm{\gamma }$ are acute angles and $\cos \mathrm{\theta }=\sin \mathrm{\beta }/\sin \mathrm{\alpha },\cos \mathrm{\varphi }=\sin \mathrm{\gamma }/\sin \mathrm{\alpha }\text{ and }\cos (\mathrm{\theta }-\mathrm{\varphi })=\sin \mathrm{\beta sin}\mathrm{\gamma }$ then the value of ${\tan }^2\mathrm{\alpha }-{\tan }^2\mathrm{\beta }-{\tan }^2\mathrm{\gamma }$ is equal to

 From the third relation we get 

$\begin{array}{l}\cos \mathrm{\theta cos}\mathrm{\varphi }+\sin \mathrm{\theta sin}\mathrm{\varphi }=\sin \mathrm{\beta sin}\mathrm{\gamma }\\ \Rightarrow {\sin }^2{\mathrm{\theta sin}}^2\mathrm{\varphi }=\left(\cos \mathrm{\theta cos}\mathrm{\varphi }-\sin \mathrm{\beta sin}\mathrm{\gamma }{)}^2\right.\\ \Rightarrow \left(1-\frac{{\sin }^2\mathrm{\beta }}{{\sin }^2\mathrm{\alpha }}\right)\left(1-\frac{{\sin }^2\mathrm{\gamma }}{{\sin }^2\mathrm{\alpha }}\right)\\ ={\left(\frac{\sin \mathrm{\beta sin}\mathrm{\gamma }}{{\sin }^2\mathrm{\alpha }}-\sin \mathrm{\beta sin}\mathrm{\gamma }\right)}^2\end{array}$

$\begin{array}{l}\Rightarrow \left({\sin }^2\mathrm{\alpha }-{\sin }^2\mathrm{\beta }\right)\left({\sin }^2\mathrm{\alpha }-{\sin }^2\mathrm{\gamma }\right)={\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }{\left(1-{\sin }^2\mathrm{\alpha }\right)}^2\\ \Rightarrow {\sin }^4\mathrm{\alpha }\left(1-{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }\right)-{\sin }^2\mathrm{\alpha }\\ \left({\sin }^2\mathrm{\beta }+{\sin }^2\mathrm{\gamma }-2{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }\right)=0\\ \therefore {\sin }^2\mathrm{\alpha }=\frac{{\sin }^2\mathrm{\beta }+{\sin }^2\mathrm{\gamma }-2{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}{1-{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}\\ \text{ and }{\cos }^2\mathrm{\alpha }=\frac{1-{\sin }^2\mathrm{\beta }-{\sin }^2\mathrm{\gamma }+{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}{1-{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}\\ \Rightarrow {\tan }^2\mathrm{\alpha }=\frac{{\sin }^2\mathrm{\beta }-{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }+{\sin }^2\mathrm{\gamma }-{\sin }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}{{\cos }^2\mathrm{\beta }-{\sin }^2\mathrm{\gamma }\left(1-{\sin }^2\mathrm{\beta }\right)}\\ =\frac{{\sin }^2{\mathrm{\beta cos}}^2\mathrm{\gamma }+{\cos }^2{\mathrm{\beta sin}}^2\mathrm{\gamma }}{{\cos }^2{\mathrm{\beta cos}}^2\mathrm{\gamma }}\\ ={\tan }^2\mathrm{\beta }+{\tan }^2\mathrm{\gamma }\\ \Rightarrow {\tan }^2\mathrm{\alpha }-{\tan }^2\mathrm{\beta }-{\tan }^2\mathrm{\gamma }=0\end{array}$