\(A=\dfrac{sin^22^0+sin^24^0+...+sin^288^0}{2\cdot tan15\cdot tan25\cdot...\cdot tan75}\)
\(=\dfrac{\left(sin^22^0+sin^288^0\right)+\left(sin^24^0+sin^286^0\right)+...+\left(sin^244^0+sin^246^0\right)}{2\cdot\left(tan15\cdot tan75\right)\cdot\left(tan25\cdot tan65\right)\left(tan35\cdot tan55\right)\cdot tan45}\)
\(=\dfrac{1+1+...+1}{2}\)
\(=\dfrac{22}{2}=11\)