Let $\gamma$ be a differentiable and regular curve in $\mathbb{R}^3$ which satisfies $|\gamma|=1$. Prove that for every point $$\kappa_1=1+\kappa_2^2$$ where $\kappa_1$ is the curvature of $\gamma$ in $\mathbb{R}^3$ and $\kappa_2$ is the geodesic curvature of $\gamma$ on the unit sphere.
I couldn't find any connection between these two properties (maybe because for most of the curves, like $\gamma(x,y,z)=\text{id}=(x,y,z)$ I can't imagine how their geodesic curvature can be on the unit sphere). I'd like to get a hint towards the solution.
See Do Carmo, http://www.maths.ed.ac.uk/~aar/papers/docarmo.pdf p. 249, we have $k_1²=k_n²+k_2²$ where $k_n$ is a normal curvature and note that in $\mathbb{R}³$, $k_n$ equals to 1.