Your code does not run because of syntax errors. I tried to restore it:
mat = {{(-2 + x) Cos[\[Pi] (-2 + x)] -
f (-2 + x) Sin[\[Pi] (-2 + x)], -f (-2 +
x) Cos[\[Pi] (-2 + x)] + (2 - x) Sin[\[Pi] (-2 + x)],
x Cos[\[Pi] x] - f (-2 + x) Sin[\[Pi] x],
-f (-2 + x) Cos[\[Pi] x] - x Sin[\[Pi] x], 0, 0, 0,
0}, {(-2 + x) Cos[\[Pi] (-2 + x)], (2 - x) Sin[\[Pi] (-2 + x)],
x Cos[\[Pi] x], -x Sin[\[Pi] x], (2 - x) Cos[\[Pi] (-2 + x)],
(2 -
x) Sin[\[Pi] (-2 +
x)], -x Cos[\[Pi] x], -x Sin[\[Pi] x]}, {1/
2 (-2 + a c - b c) (-2 + x) Cos[\[Pi] (-2 + x)],
1/2 (2 - a c + b c) (-2 + x) Sin[\[Pi] (-2 + x)],
1/2 (-2 + a c - b c) (-1 - (-2 + c - b c)/(2 - a c + b c) +
x) Cos[\[Pi] x],
1/2 (2 - a c + b c) (-1 - (-2 + c - b c)/(2 - a c + b c) +
x) Sin[\[Pi] x], (-2 + x) Cos[\[Pi] (-2 + x)],
(-2 + x) Sin[\[Pi] (-2 + x)],
(-1 + 1/2 (2 - c - a c) + x) Cos[\[Pi] x],
(-1 + 1/2 (2 - c - a c) + x) Sin[\[Pi] x]},
{Sin[\[Pi] (-2 + x)], Cos[\[Pi] (-2 + x)], Sin[\[Pi] x],
Cos[\[Pi] x], Sin[\[Pi] (-2 + x)], -Cos[\[Pi] (-2 + x)],
Sin[\[Pi] x], -Cos[\[Pi] x]}, {0, 1, 0, 1, 0, -1, 0, -1},
{-2 + x, 0, x, 0, 2 - x, 0, -x, 0},
{0, 1/2 (2 - a c + b c) (-2 + x), 0,
1/2 (2 - a c + b c) (-1 + (-2 + c - b c)/(2 - a c + b c) +
x),
0, 2 - x, 0, 1 + 1/2 (2 - c - a c) - x},
{1/2 (2 - a c + b c) (-2 + x), 0,
1/2 (2 - a c + b c) (-1 - (-2 + c - b c)/(2 - a c + b c) +
x),
0, 2 - x, 0, 1 + 1/2 (-2 + c + a c) - x, 0}};
Det[mat] // Simplify // Factor
sols = Solve[% == 0 && c > 0 && -1/2 < b < 1/2 && f != 0 &&
0 < x < 1 && c > 0, Reals]
mat2 = FullSimplify[
mat /. sols[[1]], -1/2 < b < 1/2 && f != 0 && 0 < x < 1 && c > 0]
Det[mat2] // Simplify
NullSpace[mat2]