The Sun and Moon in the first quarter phase simultaneously set over the horizon. What latitude is the observer at?

The Sun and Moon in the first quarter phase simultaneously set over the horizon. What latitude is the observer at? Ignore the refraction and parallax of the moon

The sun is always on the ecliptic. The moon in the first quarter is located 90 ° east of the sun. In this case, the Moon can be spaced from the ecliptic by an angle not exceeding the inclination of the lunar ecliptic i (5.15 °). It turns out that at the time of observation, the ecliptic is located between the two dotted lines in the figure. Consequently, one of the poles of the ecliptic (north or south) is located near the zenith, the distance to it at an angle of no more than i. In this case, the direction from the zenith to the pole of the ecliptic can be any, since the problem statement does not say where the Sun is located on the horizon. If it sets exactly to the west, then the pole of the ecliptic is removed from the zenith to the north or south. The declination of the ecliptic pole is δ = ± (90 ° – ε) = ± 66.55 °.

Here ε is the angle of inclination of the equator to the ecliptic. The latitude of the site can be equal to δ – i ≤ ϕ ≤ δ + i. As a result, we get intervals for latitude: [–71.7 °; –61.4 °]; [+ 61.4 °; + 71.7 °].