Whenever I try to find the solution to this math problem, I get two
different answers and they are both different from the answer in the
book.  Please show all the steps to getting the correct solution. 
Thanks.  Here's the problem:
From the top of a building, the angles of deppresion of two places due
west of it are sixty-three degrees and fifty-six degrees,
respectively.  Given that the foot of the building and the two places
are on ground level, and the distance between the two places is eighty
meters, find the height of the building.

Hello.

The height of the building is 484.898 meters.


The key formula here is:

                            height
tan(angle of depression) = --------
                           distance


Or expressed another way:

                 height
distance = --------------------
           tan(angle of depression)


You said that from the top of the building, the angles of depression
of the two places due west are 63 degrees and 56 degrees.  Let's call
those places PointA and PointB, respectively.


                      height of building
distance to PointA = --------------------
                       tan(63 degrees)



                      height of building
distance to PointB = --------------------
                       tan(56 degrees)


The kicker here is that we know that the distance between the two
places is 80 meteres.

In other words, distance to PointB is 80 meters more than distance to
PointA (note that PointA is the closer point because the angle of
depression is greater).  So...

distance to PointB = distance to PointA + 80 meters


Thus, we can express everything in terms of distance to PointA

                             height of building
distance to PointA + 80 m = --------------------
                              tan(56 degrees)



                      height of building
distance to PointA = --------------------
                       tan(63 degrees)


Using substitution, we can combine the two statements:


 height of building                height of building
 --------------------  + 80 m  =  --------------------
  tan(63 degrees)                   tan(56 degrees)



Or...


         height of building        height of building
80 m =  --------------------  -   --------------------
         tan(56 degrees)            tan(63 degrees)


Or 


80 m = height ( 1/tan(56 degrees) - 1/(tan(63 degree)


Or 

height = 80 m / ( 1/tan(56 degrees) - 1/(tan(63 degrees))

Getting out our calculator... Or using the Google calculator...
://www.google.com/search?hl=en&lr=&ie=ISO-8859-1&q=80%2F%28%281%2Ftan%2856+degrees%29%29+-+%281%2Ftan%2863+degrees%29%29%29&btnG=Search

We see that the height of the building is 484.898 meters.

-------
Further information:

Your problem is similar to various problems on these pages:

Trigonometry Heights and Distances:
http://acharya.iitm.ac.in/mirrors/vv/vidya/etrhtdt.html
Right Triangles
http://aleph0.clarku.edu/~djoyce/java/trig/right.html
MathUSee.com Lesson 6 Angles of Elevation and Depression
http://www.mathusee.com/pdfs/trigonometrysample.pdf

-------
search strategy:
"angle of depression" height distance

I hope this helps.

Thanks so very much for helping me out.  I really appreciate that you
showed all the steps.