A man, whose eyes are 2m above the ground , measure the angle to the top of the Eiffel tower from two positions that are 200m apart?

Put height of tower from man's horizontal line of sight = h. If a plumb line were dropped from the
top of the tower, it would cut the man's line of sight at some point D, say. If we label point of
observation where 65deg was recorded P, then d = DP. All linear measurements are in [m]. h/d = tan(65). h/(d+200) =tan(42.6). (d+200)tan(42.6) =dtan(65). d = 200tan(42.6)/[tan(65) -tan(42.6)].
h = dtan(65) = 200tan(65)tan(42.6)/[tan(65) - tan(42.6)].
= 200AB/(A-B), where A = tan(65), B = tan(42.6).
= 321.9657052 approx = 322. Now we must add on 2 to account for the height above ground
where the measurements were take. Therefore height of tower = approx 324[m].

Let's call the bigger hypoteneuse of the two triangles x and the height of the tower h.

Let's first find the angles. The supplementary angle to the 65 degrees is 180-65=115 degrees.

That makes the third angle of the triangle whose base is 200m and hypotenuse x equal to 180 - (42.6+115) = 22.4 degrees.

We now have an angle and opposite length so we can find the length we called x.

200/sin 22.4 degrees = x/sin 115degrees. That means x = (200 sin115 )/sin22.4 = 475.66 m.
Now we have the hypotenuse and opposite angle for the large right angle triangle where the height of the tower (h) is the opposite side.

h = 475.66m * sin 42.6 degrees = 322 m.

Note his eyes are 2m off the ground so the height of the tower is 322m + 2m = 324 meters.

Let h be the height of the Eiffel tower
tan(42) = h / (200+x)
200+x = h / tan(42)
200 + x = 1.1106h ----(1)

tan(65) = h/x
x = h/ tan(65)
x = 0.4663h

200 + 0.4663h = 1.1106h
200 = 0.6443h
h = 200 /0.6443 = 310.41 m

https://gyazo.com/f7004a79f76eb4194b1038cd783b8982

The angles inside the triangle for which 200m is a side
measure 42.6, (180 - 65) = 115, and (180 - 115 - 42.6) = 22.4º
Use law of sines to get the length x of the segment from the 65º angle to the top of the tower

x / sin 42.6 = 200 / sin 22.4

Once you get that, find the height of the tower above eye level h by soh cah toa; sin 65 = h/x
then add 2