solve 2arccos(x)=arccos(2x-1)

Take the cosine of both sides to
get

cos(2arccos(x))=cos(arccos(2x-1))

cos(2arccos(x))=2(cos(arccos(x)))^2-1 
and cos(arccos(2x-1))=2x-1 and
cos(arccos(x))=x so we
get

2x^2 - 1 = 2x - 1

2x^2 -
2x = 0

Factoring we
get

2x(x-1) = 0

So x = 0 or x
= 1 are our solutions.  You should substitute into the original equation to check the
answers.

2arccos(0) = arccos(-1)  which arccos(0) = pi/2
and arccos(-1) = pi so this checks.

2arccos(1) = arccos(1)
since arccos(1) = 0 this checks.

So the answers are x = 0
and x = 1.