Cho \(D = {\tan ^2}\frac{\pi }{8}.\tan \frac{{3\pi }}{8}.\tan \frac{{5\pi }}{8}\). Chọn đáp án đúng.
A. \(D = - 1\)
B. \(D = 1\)
C. \(D = \frac{1}{2}\)
D. \(D = 0\)
Sử dụng công thức: \(\tan \left( {\frac{\pi }{2} - \alpha } \right) = \cot \alpha \).
\(D = - \left( {\tan \frac{\pi }{8}.\tan \frac{{3\pi }}{8}} \right).\left[ {\tan \left( { - \frac{\pi }{8}} \right)\tan \frac{{5\pi }}{8}} \right]\)
Mà \(\frac{\pi }{8} + \frac{{3\pi }}{8} = \frac{\pi }{2},{\mkern 1mu} - \frac{\pi }{8} + \frac{{5\pi }}{8} = \frac{\pi }{2} \Rightarrow \tan \frac{{3\pi }}{8} = \cot \frac{\pi }{8},{\mkern 1mu} \tan \frac{{5\pi }}{8} = \cot \left( { - \frac{\pi }{8}} \right)\)
Nên \(D = - \left( {\tan \frac{\pi }{8}.\cot \frac{\pi }{8}} \right).\left[ {\tan \left( { - \frac{\pi }{8}} \right)\cot \left( { - \frac{\pi }{8}} \right)} \right] = - 1\)
Đáp án A
