Giá trị của biểu thức \(P = \frac{{\sin \frac{\pi }{9} + \sin \frac{{5\pi }}{9}}}{{\cos \frac{\pi }{9} + \cos \frac{{5\pi }}{9}}}\) bằng:

A. \(\frac{1}{{\sqrt 3 }}\).

B. \( - \frac{1}{{\sqrt 3 }}\).

C. \(\sqrt 3 \).

Đáp án đúng là: C

Ta có \(P = \frac{{\sin \frac{\pi }{9} + \sin \frac{{5\pi }}{9}}}{{\cos \frac{\pi }{9} + \cos \frac{{5\pi }}{9}}}\)\( = \frac{{2\sin \frac{{\frac{\pi }{9} + \frac{{5\pi }}{9}}}{2}\cos \frac{{\frac{\pi }{9} - \frac{{5\pi }}{9}}}{2}}}{{2\cos \frac{{\frac{\pi }{9} + \frac{{5\pi }}{9}}}{2}\cos \frac{{\frac{\pi }{9} - \frac{{5\pi }}{9}}}{2}}}\)

\( = \frac{{\sin \frac{\pi }{3}\cos \left( { - \frac{{2\pi }}{9}} \right)}}{{\cos \frac{\pi }{3}\cos \left( { - \frac{{2\pi }}{9}} \right)}} = \frac{{\sin \frac{\pi }{3}}}{{\cos \frac{\pi }{3}}} = \frac{{\frac{{\sqrt 3 }}{2}}}{{\frac{1}{2}}} = \sqrt 3 \).