Let's begin by simplifying each of the terms in the expression on the left-hand side of the equation:
(x^m * x^n)^(m-n) = x^(m-n)*(m-n)
(x^p * x^(-n))^(n+p) = x^(n+p)*(p-n)
(x^p * x^(-m))^(p+m) = x^(p+m)*(p-m)
Now, let's substitute the given equation p^2 - n^2 = 1 into these expressions:
x^(m-n)*(p^2-n^2) = x^(m-n)
x^(n+p)*(p^2-n^2) = x^(n+p)
x^(p+m)*(p^2-n^2) = x^(p+m)
Simplifying each of these expressions using the given equation, we get:
x^(m-n) = x^(m-n)
x^(n+p) = x^(n+p)
x^(p+m) = x^(p+m)
Therefore, the left-hand side of the equation simplifies to:
x^(m-n) * x^(n+p) * x^(p+m) = x^(m-n + n+p + p+m) = x^(2m) = x^2^(m-n + n+p + p+m)
Thus, we have proved that:
(x^m * x^n)^(m-n) * (x^p * x^(-n))^(n+p) * (x^p * x^(-m))^(p+m) = x^2
which is the same as the given equation.