Functions

Let f : R \(\to\) R be defined as \(f(x + y) + f(x - y) = 2f(x)f(y),f\left( {{1 \over 2}} \right) = - 1\). Then, the value of \(\sum\limits_{k = 1}^{20} {{1 \over {\sin (k)\sin (k + f(k))}}}\) is equal to :