若 $\int_{-\pi}^{\pi}\left(x - a_{1}\cos x - b_{1}\sin x\right)^{2}\mathrm{d}x = \min_{a,b\in \mathbf{R}}\left\{\int_{-\pi}^{\pi}\left(x - a\cos x - b\sin x\right)^{2}\mathrm{d}x\right\}$ , 则 $a_{1}\cos x + b_{1}\sin x = (\quad)$ (A) $2\sin x$ (B) $2\cos x$ (C) $2\pi \sin x$ (D) $2\pi \cos x$