$\frac{1}{1 - a} + \frac{1}{1 + a} + \frac{2}{1 + a^2} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{1 + a + 1 - a}{(1 - a)(1 + a)} + \frac{2}{1 + a^2} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{2}{1 - a^2} + \frac{2}{1 + a^2} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{2(1 + a^2) + 2(1 - a^2)}{(1 - a^2)(1 + a^2)} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{2 + 2a^2 + 2 - 2a^2}{(1 - a^2)(1 + a^2)} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{4}{1 - a^4} + \frac{4}{1 + a^4} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{4(1 + a^4) + 4(1 - a^4)}{(1 - a^4)(1 + a^4)} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{4 + 4a^4 + 4 - 4a^4}{(1 - a^4)(1 + a^4)} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{8}{1 - a^8} + \frac{8}{1 + a^8} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{8(1 + a^8) + 8(1 - a^8)}{(1 - a^8)(1 + a^8)} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{8 + 8a^8 + 8 - 8a^8}{(1 - a^8)(1 + a^8)} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{16}{1 - a^{16}} + \frac{16}{1 + a^{16}} = \frac{32}{1 - a^{32}}$
$\frac{16(1 + a^{16}) + 16(1 - a^{16})}{(1 - a^{16})(1 + a^{16})} = \frac{32}{1 - a^{32}}$
$\frac{16 + 16a^{16} + 16 - 16a^{16}}{(1 - a^{16})(1 + a^{16})} = \frac{32}{1 - a^{32}}$
$\frac{32}{1 - a^{32}} = \frac{32}{1 - a^{32}}$