To solve this trigonometric equation, we can use the cosine addition and subtraction identities:

cos$A$ - cos$B$ = -2sin$(A+B)/2$sin$(A-B)/2$ cos$A$ + cos$B$ = 2cos$(A+B)/2$cos$(A-B)/2$

Given equation:
cos$9x$ - cos$7x$ + cos$3x$ - cos$x$ = 0

Apply the cosine subtraction identity to simplify the equation:
-2sin$(9x+7x)/2$sin$(9x-7x)/2$ + 2sin$(3x+x)/2$sin$(3x-x)/2$ = 0
-2sin$8x$sin$x$ + 2sin$2x$sin$x$ = 0

Factor out sin$x$:
-2sin$x$$$sin(8x)-sin(2x)$$ = 0

Set each factor equal to zero to find the solutions:
sin$x$ = 0
This gives x = 0, π, 2π, ...

sin$8x$ - sin$2x$ = 0
sin$8x$ = sin$2x$

Since sin$A$ = sin$B$ when A = nπ + $-1$^n*B $nisaninteger$, we solve for x:
8x = 2x + n$2\pi$ 6x = n$2\pi$ x = n$\pi /3$

Therefore, the solutions to the equation cos$9x$ - cos$7x$ + cos$3x$ - cos$x$ = 0 are:
x = 0, π, 2π, ..., n$\pi /3$ where n is an integer.