$\frac{16}{(a - 2)^4} : (\frac{1}{(a - 2)^2} - \frac{2}{(a^2 - 4)} + \frac{1}{(a + 2)^2}) - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} : (\frac{1}{(a - 2)^2} - \frac{2}{(a - 2)(a + 2)} + \frac{1}{(a + 2)^2}) - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} : \frac{(a + 2)^2 - 2(a - 2)(a + 2) + (a - 2)^2}{(a - 2)^2(a + 2)^2} - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} : \frac{(a + 2 - (a - 2))^2}{(a - 2)^2(a + 2)^2} - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} : \frac{(a + 2 - a + 2)^2}{(a - 2)^2(a + 2)^2} - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} : \frac{4^2}{(a - 2)^2(a + 2)^2} - \frac{8a}{(a - 2)^2} = 1$
$\frac{16}{(a - 2)^4} * \frac{(a - 2)^2(a + 2)^2}{16} - \frac{8a}{(a - 2)^2} = 1$
$\frac{1}{(a - 2)^2} * \frac{(a + 2)^2}{1} - \frac{8a}{(a - 2)^2} = 1$
$\frac{(a + 2)^2 - 8a}{(a - 2)^2} = 1$
$\frac{a^2 + 4a + 4 - 8a}{(a - 2)^2} = 1$
$\frac{a^2 - 4a + 4}{(a - 2)^2} = 1$
$\frac{(a - 2)^2}{(a - 2)^2} = 1$
1 = 1

$\frac{a + 11}{a + 9} - (\frac{a + 5}{a^2 - 81} + \frac{a + 7}{a^2 - 18a + 81}) : (\frac{a + 3}{a - 9})^2 = 1$
$\frac{a + 11}{a + 9} - (\frac{a + 5}{(a - 9)(a + 9)} + \frac{a + 7}{(a - 9)^2}) : \frac{(a + 3)^2}{(a - 9)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{(a + 5)(a - 9) + (a + 7)(a + 9)}{(a - 9)^2(a + 9)} * \frac{(a - 9)^2}{(a + 3)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{a^2 + 5a - 9a - 45 + a^2 + 7a + 9a + 63}{a + 9} * \frac{1}{(a + 3)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{2a^2 + 12a + 18}{a + 9} * \frac{1}{(a + 3)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{2(a^2 + 6a + 9)}{a + 9} * \frac{1}{(a + 3)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{2(a + 3)^2}{a + 9} * \frac{1}{(a + 3)^2} = 1$
$\frac{a + 11}{a + 9} - \frac{2}{a + 9} = 1$
$\frac{a + 11 - 2}{a + 9} = 1$
$\frac{a + 9}{a + 9} = 1$
1 = 1