Let $u,v,x \in R^d$ be three unit vectors.
I found a very complicated proof that $(v^Tx)^2-(u^Tx)^2 \leq 1-(u^Tv)^2$. That is $||uu^T-vv^T||_2 = 1-(u^Tv)^2$, or that $f(v,x)\leq f(v,u)+f(u,x)$ where $f(v,u)=1-(v^Tu)^2$ is the squared sin of the angle between $u$ and $v$.

Is there a one-liner proof?

Say, using the (spherical?) law of cosine or the Haversine formula? induced norm for positive semi-definite matrices?