This function is one-to-one.
Work Step by Step
The cosine function is continuous and decreasing when $x\in[0,\pi]$. This means that $h(x)=1+\cos x$ is also continuous and decreasing on $[0,\pi]$. So, for every $x_1,x_2\in[0,\pi]$ (which is the domain) $x_1h(x_2)$ and if $x_2$ is greater then $h(x_2)>h(x_1)$.