the sun - How to get the longitude/latitude from solar zenith/azimuth?

Well let's see:

Local Standard Time Meridian (LSTM)

The Local Standard Time Meridian (LSTM) is a reference meridian used for a particular time zone and is similar to the Prime Meridian, which is used for Greenwich Mean Time.
The (LSTM) is calculated according to the equation:
$$
LSTM = 15^{o}.Delta T_{GMT}
$$
where $Delta T_{GMT}$ is the difference of the Local Time (LT) from Greenwich Mean Time (GMT) in hours.

Equation of Time (EoT)

The equation of time (EoT) (in minutes) is an empirical equation that corrects for the eccentricity of the Earth's orbit and the Earth's axial tilt.
$$
EoT = 9.87 sinleft(2Bright) - 7.53cosleft(Bright) - 1.5 sinleft(Bright)
$$

Where $B = frac{360}{365}left(d - 81right) $ in degrees and d is the number of days since the start of the year.

Time Correction Factor (TC)

The net Time Correction Factor (in minutes) accounts for the variation of the Local Solar Time (LST) within a given time zone due to the longitude variations within the time zone and also incorporates the EoT above.
$$
TC = 4left(Longitude - LSTMright) + EoT
$$
The factor of 4 minutes comes from the fact that the Earth rotates 1° every 4 minutes.

Local Solar Time (LST)

The Local Solar Time (LST) can be found by using the previous two corrections to adjust the local time (LT).
$$
LST = LT + frac{TC}{60}
$$

Hour Angle (HRA)

The Hour Angle converts the local solar time (LST) into the number of degrees which the sun moves across the sky. By definition, the Hour Angle is 0° at solar noon. Since the Earth rotates 15° per hour, each hour away from solar noon corresponds to an angular motion of the sun in the sky of 15°. In the morning the hour angle is negative, in the afternoon the hour angle is positive.
$$
HRA = 15^{o}left(LST - 12right)
$$

Declination angle:

The declination angle denoted by $delta$, varies seasonally due to the tilt of the Earth on its axis of rotation and the rotation of the Earth around the sun. If the Earth were not tilted on its axis of rotation, the declination would always be 0°. However, the Earth is tilted by 23.45° and the declination angle varies plus or minus this amount. Only at the spring and fall equinoxes is the declination angle equal to 0°.
$$
delta = 23.45^{o} sinleft[frac{360}{365}left(d - 81right)right]
$$

where d is the day of the year with Jan 1 as d = 1.

Elevation angle:

The elevation angle is the angular height of the sun in the sky measured from the horizontal.
$$
α=sin^{−1}[sindelta sinphi + cosdelta cosphi cos(HRA)]
$$
Where $delta$ is the declination angle, $phi$ is the local latitude and HRA is the Hour angle.

The azimuth angle is the compass direction from which the sunlight is coming. At solar noon, the sun is always directly south in the northern hemisphere and directly north in the southern hemisphere.
$$
Azimuth = cos^{-1}left[frac{sindelta cosphi - cosdelta senphi cos(HRA)}{cosalpha}right]
$$

Where $delta$ is the declination angle, $phi$ is the local latitude and HRA is the Hour angle.

The zenith angle is the angle between the sun and the vertical. The zenith angle is similar to the elevation angle but it is measured from the vertical rather than from the horizontal, thus making the zenith
$$
Zenith = 90° - alpha
$$

Where $alpha$ is the elevation angle.

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Note that your input parameters are going to be:

• Longitude


• $Delta T_{GMT}$ is the difference of the Local Time (LT) from Greenwich Mean Time (GMT) in hours


• LT local military time in hours


• $phi$ the local latitude


• d the day of the year