Solve Delta_{1}={ccc}{1}&{cosalpha}&{cosbeta}{cosalpha}&{1}&{cosgamma}{cosbeta}&{cosalpha}&{1}

$$\Delta _ { 1 } = \left| \begin{array} { c c c } { 1 } & { \cos \alpha } & { \cos \beta } \\ { \cos \alpha } & { 1 } & { \cos \gamma } \\ { \cos \beta } & { \cos \alpha } & { 1 } \end{array} \right|$$

Assign Δ_1

$$\Delta _{1}≔\cos(\alpha )\cos(\beta )\cos(\gamma )+\cos(\beta )\left(\cos(\alpha )\right)^{2}-\cos(\alpha )\cos(\gamma )-\left(\cos(\alpha )\right)^{2}-\left(\cos(\beta )\right)^{2}+1$$