a.$m^3=(tana+cota)^3=tan^3a+cot^3a+3tana.cota(tana+cota)$
$\Rightarrow m^3=tan^3a+cot^3a+3m$
$\Rightarrow tan^3a+cot^3a=m^3-3m$
$m^2=(tana+cota)^2=tan^2a+cot^2a+2tana.cota$
$\Rightarrow tan^2a+cot^2a=m^2-2$
$(tana-cota)^2=tan^2a+cot^2a-2tana.cota=m^2-4$
$\Rightarrow |tana=cota|=\sqrt{m^2-4}$
b.
$\frac{2sina+3cosa}{4sina-5cosa}=\frac{cosa(2tana+3)}{cosa(4tana-5)}=\frac{2.3+3}{4.3-5}=\frac{9}{7}$
$\frac{3sina-2cosa}{5sin^3a+4cos^3a}$
$=\frac{cosa(3tana-2)}{cos^3a(5tan^3a+4)}=\frac{1}{cos^2a}.\frac{3.3-2}{5.3^3+4}=\frac{1}{cos^2a}.\frac{7}{139}$
Vì $tana=3\Rightarrow sina=3cosa\Rightarrow sin^2a=9cos^2a$
Mà $sin^a+cos^2a=1\Rightarrow cos^2a=\frac{1}{10}$
$\Rightarrow \frac{3sina-2cosa}{5sin^3a+4cos^3a}=\frac{70}{139}$