Cho \(a + b + c = \frac{\pi...

C

- Hướng dẫn giải

Ta có

\(\begin{array}{l}a + b + c = \frac{\pi }{2} \Leftrightarrow a + b = \frac{\pi }{2} \Rightarrow \cot \left( {a + b} \right) = \cot \left( {\frac{\pi }{2} - c} \right) = \tan c \Rightarrow \frac{{\cot {\rm{a}}.\cot b - 1}}{{\cot {\rm{a}} + \cot b}} = \frac{1}{{\cot c}}\\a + b + c = \frac{\pi }{2} \Leftrightarrow a + b = \frac{\pi }{2} \Rightarrow \cot \left( {a + b} \right) = \cot \left( {\frac{\pi }{2} - c} \right) = \tan c \Rightarrow \frac{{\cot {\rm{a}}.\cot b - 1}}{{\cot {\rm{a}} + \cot b}} = \frac{1}{{\cot c}}\\ \Leftrightarrow \cot {\rm{a}}.\cot b.\cot c = \cot {\rm{a}} + \cot b + \cot c\end{array}\)

Mà \(\cot {\rm{a}} + \cot c = 2\cot b\)

Do đó ta được \(\cot {\rm{a}}.\cot b.\cot c = 3\cot b \Rightarrow \cot {\rm{a}}.\cot c = 3\)