Question

À boy observés that the angle of élévation of the top of the tower is 32degree and the boy moves 8m to the tower and sees the angle of élévation is 43 find the height of the tower

Answer 1

The traditional way to do this style of question has you ending up with
height = 8*tan32*tan43/(tan43 - tan32) = ....

I found through the years that students follow more readily the following method:

Label the top of the tower P and its bottom Q
label the boy's original position A and his new position B
It is easy find all the angles in triangle ABP.
then in triangle ABP:
BP/sin32° = 8/sin11°
BP = 8sin32/sin11

in the right-angled triangle BPQ
sin 43° = h/BP
h = BPsin43 = 8sin32sin43/sin11 = .... for the same answer

Answer 2

or, as a slightly less opaque version of the first solution, if the height is h, then
h cot32° - h cot43° = 8
But you do have to be comfortable thinking in terms of cotangent, rather than tangent.

Answer 3

X = hor. distance to base of tower.
x-8 = 8 m close.

Tan32 = h/x.
h = x*Tan32.

Tan43 = h/(x-8)
h = (x-8)*Tan43.

h = x*Tan32 = (x-8)*Tan43
0.62x = (x-8)0.93
X = 24 m.

h = x*Tan32 = 24*0.62 = 15 m.