Zeros of a Cosine Hybrid

The function $f$ is of the form $f(x) = \dfrac{k}{x}$, and the curve $g(x) = f(x)\cos x + b$ has tangent line $y = -\dfrac{6}{\pi}x + 2$ at $x = \dfrac{\pi}{2}$.

1. Find $g(x)$. 2. Determine, with proof, the number of zeros of $F(x) = g(x) + 1 - \dfrac{3}{2\pi}$ on $(0, 2\pi]$.