2. Given the function $f(x)=\left(\frac{1}{a^{x}-1}+\frac{1}{2}\right) \cos x$ $+b \sin x+6$ (where $a$ and $b$ are constants, and $a>1$), $f\left(\lg \log _{8} 1000\right)=8$. Then the value of $f(\lg \lg 2)$ is ( ).
(A) 8
(B) 4
(C) -4
(D) depends on the values of $a$ and $b$