(a) Rút gọn M = tan 10 ∘ ⋅ tan 20 ∘ ⋅ tan 30 ∘ ⋅ tan 40 ∘ ⋅ tan 50 ∘ ⋅ tan 60 ∘ ⋅ tan 70 ∘ ⋅ tan 80 ∘ . (b) Cho π/2 < α < π . Xác định dấu của biểu thức tan ( 3 π/2 − α ) .

a) \(M = \tan 10^\circ \cdot \tan 20^\circ \cdot \tan 30^\circ \cdot \tan 40^\circ \cdot \tan 50^\circ \cdot \tan 60^\circ \cdot \tan 70^\circ \cdot \tan 80^\circ \)

\[M = \left( {\tan 10^\circ \cdot \tan 80^\circ } \right) \cdot \left( {\tan 20^\circ \cdot \tan 70^\circ } \right) \cdot \left( {\tan 30^\circ \cdot \tan 60^\circ } \right) \cdot \left( {\tan 40^\circ \cdot \tan 50^\circ } \right)\]

\[M = \left( {\tan 10^\circ \cdot \cot 10^\circ } \right) \cdot \left( {\tan 20^\circ \cdot \cot 20^\circ } \right) \cdot \left( {\tan 30^\circ \cdot \cot 30^\circ } \right) \cdot \left( {\tan 40^\circ \cdot \cot 40^\circ } \right)\]

\[M = 1 \cdot 1 \cdot 1 \cdot 1 = 1\].

b) \(\tan \left( {\frac{{3\pi }}{2} - \alpha } \right)\)\( = \tan \left( {\pi + \frac{\pi }{2} - \alpha } \right)\)\( = \tan \left( {\frac{\pi }{2} - \alpha } \right)\)\( = \cot \alpha \).

Mà \(\frac{\pi }{2} < \alpha < \pi \) nên \(\left. \begin{array}{l}\sin \alpha > 0\\\cos \alpha < 0\end{array} \right\} \Rightarrow \cot \alpha < 0\).

Vậy với \(\frac{\pi }{2} < \alpha < \pi \) thì \(\tan \left( {\frac{{3\pi }}{2} - \alpha } \right) < 0\).