Limit of $(\sqrt{1-\sin^2\left(t\right)}-1+t-\sin\left(t\right),\:\sqrt{1-\sin^2\left(t\right)}-1-t+\sin\left(t\right))$ = $0$ - Is my proof good?

Question

The original limit is: $\left(\cos t-1+t-\sin t,\:\cos t-1-t+\sin t \right)$
But I changed it to get all for $\sin t$:

$\left(\sqrt{1-\sin^2 t}-1+t-\sin t,\:\sqrt{1-\sin^2 t}-1-t+\sin t\right)$

Limit of this, as $t$ goes to $0$.

I said it like that:
We will get basically $\lim_{t\to 0} \frac{(0,0)}t $ and since its $0$ by $t$, so its $0$.

Is it true? because I don't really get $0$ identity... it is super close to $0$, but not really.
Is there any way to prove it's equal to $0$? Or is my proof good?