**Toolbox:**

- If $A=\begin{vmatrix}a_{11} & a_{21} & a_{31}\\a_{12} & a_{22} & a_{32}\\a_{13} & a_{23} & a_{33}\end{vmatrix}$
- $|A|=a_{11}(a_22\times a_{33}-a_{32}\times a_{23})-a_{21}(a_{12}\times a_{33}-a_{32}\times a_{13})+a_{31}(a_{12}\times a_{23}-a_{22}\times a_{13})$

Let $\Delta=\begin{vmatrix}1 & cos C & cos B\\cos C & 1 & cos A\\cos B & cos A& 1\end{vmatrix}$

Expanding along $R_1$ we get,

$\Delta=1(1-cos^2A)-cos C(cos C-cos A.cos B)+cos B(cos C .cos A-cos B)$

$\quad=1-cos^2A-cos^2C+cos A.cos B.cos C+cos A.cos B.cos C-cos^2B.$

$\quad=1+2cos A.cos B.cos C-(cos^2A+cos^2B+cos^2C)$

But $cos^2A+cos^2B+cos^2C=1+2cos A cos B cosC.$

Substituting this we get,

$\Delta=1+2cos A cos B cos C-1-2cos A cos B cos C=0.$

Hence $\Delta=0.$