A marble statue of height h₁ metres is mounted on a pedestal. The angles of the top and bottom of the statue from a point h₂ metres above the ground level are A and B respectively..

Show that the height of the pedestal is

(h1 - h ₂ )tanB + h ₂ tanA

tanA- tan B

As you have not mentioned h₁+h > h₂ or h₁+h

h2 , Lets say the figure is :

Let the height of  pedestrial be h , or MR =h

Now , in ΔPQT

PQ= PM+MR - QR = PM + MR - TS (Since QR = TS)

PQ = h₁ + h - h₂

Also tan A = PQ/QT = (h₁ + h - h₂)  / QT

QT  = (h₁+h-h₂)/tanA  ------------ (1)

Now in ΔQTM 

tan B  = QM / QT  

QM  = QT × tan B  

QM  =  ((h₁+h-h₂)/tanA) × tan B [ From eq (1)]  ----------------(2)

Now QM = PM - PQ = h₁ - (h₁+h-h₂) = h₂-h

Thus , putting value of QM in eq (2) we get

h₂ - h = ((h₁+h-h₂)/tanA) × tan B

h₂ tanA - h tanA = h₁tanB + htanB - h₂tanB

h(tanA + tanB) = (h₂-h₁)tanB + h₂tanA

h = ((h₂-h₁)tanB + h₂tanA)/(tanA + tanB)

Similarly for other case in which h₁+h.