p. 519

87.

\(f[x_]\  := \ ArcSin[\(x - 1\)\/\(x + 1\)]; 

    g[x_]\  := \ 2  ArcTan[\@x]; \)

Plot[f[x], {x, 0, 10}]

Plot[g[x], {x, 0, 10}]

The plot of g(x) looks like the plot of f(x) shifted up. Plotting the difference indicates it is a constant equal to π/2.

Plot[g[x]  - f[x], {x, 0, 10},  PlotRange -> {0, 2}]

g[0] - f[0]

To prove that the functions differ by a constant we take their derivatives:

f'[x]

Simplify[f'[x]]

I had trouble getting Mathematica to simplify this further (the program does not know that x is a real number or x  >  0) , so I looked up Simplify[]  in the Help Browser and found some other  functions to try. PowerExpand[] did the simplification I was looking for (Of course, getting Mathematica to simplify this expression is purely optional. You can always do the derivatives and simplifications by hand.)

PowerExpand[Simplify[f'[x]]]

Now we see that g(x) has the same derivative.

g'[x]

So the two functions do indeed differ by a constant. This constant is given by g(0)  -  f(0)  =  π/2 as determined above, so g(x)  =  f(x)  +  π/2.