\(A=\frac{sin5.sin55.sin65}{cos5.cos55.cos65}=\frac{-\frac{1}{2}\left(cos60-cos50\right)sin65}{\frac{1}{2}\left(cos60+cos50\right)cos65}=\frac{-\frac{1}{2}sin65+cos50.sin65}{\frac{1}{2}cos65+cos50.cos65}\)
\(=\frac{-\frac{1}{2}sin65+\frac{1}{2}sin115+\frac{1}{2}sin15}{\frac{1}{2}cos65+\frac{1}{2}cos115+\frac{1}{2}cos15}=\frac{\frac{1}{2}sin15}{\frac{1}{2}cos15}=tan15\)
\(tan30=\frac{1}{\sqrt{3}}=tan2.15=\frac{2tan15}{1-tan^215}\Rightarrow tan^215+2\sqrt{3}tan15-1=0\)
\(\Rightarrow tan15=2-\sqrt{3}\Rightarrow A=2-\sqrt{3}\)
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