Trying to prove that this function $ F: \mathbb{R}^2 \rightarrow \mathbb{R}^2 $ is Lipschitz continuous: $ F(x,y) = (ax+\sin^2(x+y), ay +\cos^2(x+y))$ with $ a \in \mathbb{R} $
So I am trying to proove that we can find $k$ such as: $ d( f(x,y),f(t,z) \leq k d( (x,y)(t,z)) $
$ \sqrt{ [at + \sin^2(t + z) - ax - \sin^2(x + y)]^2 +[az + \cos^2(t + z)-ay - \cos^2(x + y)]^2} \\ \leq k (\sqrt{(t-x)^2 +(z-y)^2}) $
But I can't find the trick. Any idea?