Question

the angle of elevation of the top of a tower is observe from a point on the ground is 'a' and on moving a distance 'c' towards the tower the angle of elevation is 'b'.prove that the height of the tower is
$\frac{c \:( tan \: a)(tan \: b)}{tan \: b - tan \: a)}$

Answer 1

Solution:

Given:
A
I\\
l  \ \
I    \  \
I      \   \
I        \    \
I          \     \
I            \      \
L____b \__a\       (c represents angle)
B           C      D
             I--------I
                   c
(Rough diagram)
AB=height (represent as h)

          tan a = AB/BD =AB/BC+c
          tan b = AB/BC
hence,

    c(tan a)(tan b)/tan b - tan a    c(AB/BC+c)(AB/BC)
                                                   =----------------------------------
                                                            AB/BC-AB/BC+c

                                                        c(AB²/BC²+BCc)
                                                   =---------------------------------------------
                                                      AB(BC+c)-AB(BC)/(BC)(BC+c)

                                                        cAB(AB/(BC²+BCc))
                                                   =---------------------------------------------
                                                        AB(BC+c-BC)/(BC²+BC+c)
 
                                                        cAB(AB)
                                                  =---------------------
                                                        AB(c)
                                                 = cAB(AB)/ABc
                                                =AB=h=RHS hence proved
                          Hope It Helps