If  $f\left(x\right)=\left|\begin{array}{ccc}\sin x& \sin a& \sin b \\ \cos x& \cos a& \cos b\\ \tan x& \tan a& \tan b\end{array}\right|$ where  $0<a<b<\frac{\pi }{2}$ then in the interval (a, b) the function  $f^{ \text{'}}\left(x\right)=0$ has 

$f\left(a\right)=\left|\begin{array}{ccc}\sin a& \sin a& \sin b \\ \cos a& \cos a& \cos b\\ \tan a& \tan a& \tan b\end{array}\right|=0$     also $f\left(b\right)=0$

$\sin x,\cos x$ and $\tan x$ are continuous and differentiable in $\left(a,b\right)$ for $0<a<b<\frac{\pi }{2}$ by       Rolle’s theorem $\exists$ a real number $c\in \left(a,b\right)$ such that $f^{\text{'}}\left(c\right)=0$