Prove that $\|UVU^{-1}V^{-1}-I\|\leq 2\|U-I\|\|V-I\|$ $U,V$ are unitary $n\times n$ matrices, and the norm is the operator norm (so we can use $\|UV\|\leq\|U\|\|V\|$).
I've noticed that
\begin{align}
\|UVU^{-1}V^{-1}-I\|&= \|(UV-VU)U^{-1}V^{-1}\|\\
&\leq \|UV-VU\|\|U^{-1}V^{-1}\|
\end{align}
I can bound the first term by $\|UV\|+\|VU\|$, but I don't think this is useful. 
Hints (rather than complete answers) would be appreciated.
The question comes from here (exercise 1)

Let $[A,B]=AB-BA$ denote the commutator of two $n \times n$ matrices $A,B$.

Hint: For $U,V$ unitary $n\times n$ matrices, one has the identity
  $$
\lVert UVU^{-1}V^{-1}-1 \rVert = \lVert [U,V]U^{-1}V^{-1} \rVert = \lVert [U,V] \rVert = \lVert [U-1,V-1] \rVert.
$$

