Show that definite elliptic integrals $ E_1,E_2 $
$$ E_1= \int_x^ {\pi/2} \frac {dt} {\sqrt{1-k^2 \cos^2 t } }\,, \ E_2= \int_x^{\pi/2} \frac {\cos^4 t\, dt} {\sqrt{1-k^2 \cos^2 t } }\,;$$
divide to a constant quotient independent of $x$ (when $k$ constant).
Note that if $f(x)=E_{1}(x)/E_{2}(x)$ is a constant then $f'(x) =0$ ie $$E_{1}'(x)E_{2}(x)=E_{1}(x)E_{2}'(x)$$ which is the same as $$f(x) =\frac{E_{1}(x)}{E_{2}(x)}=\frac{E_{1}'(x)}{E_{2}'(x)}$$ so that the ratio of the derivatives also equals the same constant $f(x) $. But in our case $$\frac{E_{1}'(x)}{E_{2}'(x)}=\sec^{4}(x)$$ and this is clearly not a constant. I think there is some typo or other mistake in the problem.