Matrizes e Determinantes

Mostre que:
[latex]\begin{vmatrix} 1 & sena & cosa \\ 1 & senb & cosb \\ 1 & senc & cosc \end{vmatrix}[/latex], é igual a 4 . sen(b-c)/2 . sen(a-c)/2 . sen(a-b)/2

Determinante:

[latex]senAcosB - cosAsenB - senAcosC + cosAsenC + senBcosC - senCcosB = \\ \\ sen(A - B) + sen(C - A) + sen(B - C)[/latex]

Werner:
[latex]sen(A - B) + sen(C - A) + sen(B - C) = \\ \\ 2sen \dfrac{C-B}{2}cos\dfrac{2A - B - C}{2} + sen(B - C) = \\ \\ 2sen \dfrac{C-B}{2}cos\dfrac{2A - B - C}{2} - sen(C - B) = \\ \\ 2sen \dfrac{C-B}{2}cos\dfrac{2A - B - C}{2} - 2sen\dfrac{C - B}{2}cos\dfrac{C - B}{2} = \\ \\ 2sen\dfrac{C - B}{2}(cos\dfrac{2A - B - C}{2} - cos\dfrac{C - B}{2}) = \\ \\ 2sen\dfrac{C - B}{2}(-2sen\dfrac{A - B}{2}sen\dfrac{A - C}{2}) = \\ \\ -4sen\dfrac{C - B}{2}sen\dfrac{A - B}{2}sen\dfrac{A - C}{2} = \\ \\ 4sen\dfrac{B - C}{2}sen\dfrac{A - B}{2}sen\dfrac{A - C}{2}[/latex]

Vamos considerar que
\[
a > b > c, \quad a,b,c \in \mathbb{R}
\]
Desenvolvendo o determinante:
\[
\begin{align*}
\begin{vmatrix}
1 & \sin a & \cos a \\
1 & \sin b & \cos b \\
1 & \sin c & \cos c
\end{vmatrix} & = \sin b \cos c + \sin a \cos b + \sin c \cos a - ( \sin b \cos a +\sin a \cos c + \sin c \cos b) \\
& = \sin b \cos c - \sin c \cos b + \sin a \cos b - \sin b \cos a + \sin c \cos a - \sin a\cos c \\
& = \sin (b-c) + \sin (a-b) + \sin (c-a) \\
& = \sin (b-c) + \sin ( a-b) - \sin (a-c) \\
& = 2\sin \left( \frac{a-c}{2} \right)\cos \left( \frac{a-2b+c}{2} \right) - \sin (a-c) \\
& = 2\sin \left( \frac{a-c}{2} \right)\cos \left( \frac{a-2b+c}{2} \right) - 2\sin \left( \frac{a-c}{2} \right) \cos \left( \frac{a-c}{2} \right) \\
& = 2\sin \left( \frac{a-c}{2} \right) \left[ \cos \left( \frac{a-2b+c}{2} \right) - \cos \left( \frac{a-c}{2} \right)\right] \\
& = 2\sin \left( \frac{a-c}{2} \right) \cdot \left[ -2 \sin \left( \frac{a-b}{2} \right) \sin \left( \frac{c-b}{2} \right)\right] \\
& = 4 \sin \left( \frac{a-c}{2} \right) \sin \left( \frac{a-b}{2} \right) \sin \left( \frac{b-c}{2} \right)
\end{align*}
\]
Fica provado que
\[
\begin{vmatrix}
1 & \sin a & \cos a \\
1 & \sin b & \cos b \\
1 & \sin c & \cos c
\end{vmatrix} = 4 \sin \left( \frac{a-b}{2} \right)\sin \left( \frac{b-c}{2} \right)\sin \left( \frac{a-c}{2} \right) \hspace{8em} {}_{\blacksquare}
\]

Uma outra forma, aplicando a regra de Chió pra diminuir o tamanho da matriz. Não via muitas pessoas falando sobre essa "regra", então vou por ela aqui.
[latex]\begin{vmatrix} 1 & \sin\, a & \cos\, a\\ 1 & \sin\, b & \cos\, b \\ 1 & \sin\, c & \cos\, c \end{vmatrix}[/latex]

Aplicando a regra de Chió:

[latex]\\\begin{vmatrix} \sin\, b - \sin\, a & \cos\, b- \cos\, a \\ \sin\, c - \sin \, a & \cos\, c - \cos\, a \end{vmatrix}\\\\ \begin{vmatrix} 2 \cos \left( \frac{a+b}{2} \right ) \sin \left(\frac{b - a}{2} \right ) & -2\sin \left(\frac{a+b}{2} \right )\sin \left(\frac{b-a}{2} \right )\\ 2 \cos \left(\frac{a+c}{2} \right )\sin \left(\frac{c-a}{2} \right ) & -2\sin \left(\frac{c+a}{2} \right )\sin \left(\frac{c-a}{2} \right ) \end{vmatrix}\\\\ 4\sin \left(\frac{c-a}{2} \right )\sin \left(\frac{b-a}{2} \right )\left(\sin \left(\frac{a+b}{2} \right )\cos \left(\frac{a+c}{2} \right ) - \cos \left(\frac{a+b}{2} \right )\sin \left(\frac{a+c}{2} \right )\right )\\\\ 4\sin \left(\frac{a-c}{2} \right )\sin \left(\frac{a-b}{2} \right )\sin \left(\frac{b-c}{2} \right )[/latex]