arctan(x-1)+arctan(x)=pi/4
Solution :
Suppose arctan(x-1)=a
tan(a)=x-1 [1]
arctan(x)=b
tan(b)=x [2]
a+b=pi/4
a=pi/4-b [3]
from [1] and [3] :
tan(a)=tan(pi/4-b)
tan(a)=[tan(pi/4)-tan(b)]/[1+tan(pi/4)*tan(b)]
tan(a)=[1-tan(b)]/[1+tan(b)] [4]
from [1], [2], and [4] :
x-1=(1-x)/(1+x)
x-1=-(x-1)/(x+1)
x-1+(x-1)/(x+1)
(x-1)*[1+1/(x+1)]=0
x-1=0
x1=1
1+1/(x+1)=0
[x+1+1]/(x+1)=0
x+2=0
x2=-2
the final answer is x=1 or -2
