詳解：(A)  \(\tan 50^\circ \)  ＜  \(\sec 50^\circ \)

\(\sin 50^\circ \) ＜ \(1\quad  \Rightarrow \quad \frac{{\sin 50^\circ }}{{\cos 50^\circ }}\) ＜  \(\frac{1}{{\cos 50^\circ }}\)    正確

(B) \(\tan 50^\circ \) ＜ \(\cot 50^\circ \)

\(\sin 40^\circ \) ＜  \(\sin 50^\circ \quad  \Rightarrow \quad \cos 50^\circ \) ＜ \(\sin 50^\circ \)

\(\frac{{\cos 50^\circ }}{{\sin 50^\circ }}\;\) ＜ \(\;\frac{{\sin 50^\circ }}{{\cos 50^\circ }}\quad  \Rightarrow \quad \cot 50^\circ \) ＜ \(\tan 50^\circ \)  錯誤

(C) \(\sin 50^\circ \) ＜ \(\cos 50^\circ \)

\(\sin 40^\circ \) ＜ \(\sin 50^\circ \quad  \Rightarrow \quad \cos 50^\circ \) ＜ \(\sin 50^\circ \)  錯誤

(D) \(\cos 230^\circ \) ＜ \(\sin 230^\circ \)

落在第三象限 \(\cos 230^\circ  =  - \cos 50^\circ \) ， \(\sin 230^\circ  =  - \sin 50^\circ \)

\(\therefore \) \(\cos 230^\circ \) ＞ \(\sin 230^\circ \)

故 \(\cos 230^\circ \) ＜ \(\sin 230^\circ \)  錯誤