What I get:

X = 1.90661;
Y = -0.09122;
Z = -0.43421;
U = 0.43308;
V = 1.906868;
W = -0.09122;

AbsoluteTiming[
 ns2 = NSolve[{X*(Sqrt[1 - c^2]*Sqrt[1 - b^2] - Sqrt[1 - a^2]*b*c) + 
      Y*(Sqrt[1 - c^2]*b + Sqrt[1 - a^2]*Sqrt[1 - b^2]*c) + Z*(c*a) ==
      U, X*(-c*Sqrt[1 - b^2] - Sqrt[1 - a^2]*b*Sqrt[1 - c^2]) + 
      Y*(-c*b + Sqrt[1 - a^2]*Sqrt[1 - b^2]*Sqrt[1 - c^2]) + 
      Z*(Sqrt[1 - c^2]*a) == V, 
    X*(a*b) - Y*(a*Sqrt[1 - b^2]) + Z*(Sqrt[1 - a^2]) == W}, {a, b, 
    c}]]

(* Out[701]= {2.675, {{a -> -0.0000269266 - 0.000036837 I, 
   b -> -2473.26 + 3064.65 I, 
   c -> -2523.85 - 2899.76 I}, {a -> -6.50246 - 5.90991 I, 
   b -> -0.0464159 + 0.223457 I, 
   c -> -0.971354 + 0.00708998 I}, {a -> 0.127547 - 2.75549 I, 
   b -> -0.0484885 + 0.224359 I, 
   c -> -0.97638 - 0.00672336 I}, {a -> -0.0307457 - 0.167662 I, 
   b -> -0.25573 + 1.04811 I, c -> -1.33766 - 0.35285 I}}} *)

I want values of a,b,c as output.