analysis - How can I obtain this division's limit without using derivatives?

$$\lim_{y\to 0} \frac{y}{\cos\left(\frac{\pi}{2}(1+y)\right)}$$
Can anybody help me? I can use basic properties of limits, and some of those basic known limits. I know it would be easier with derivatives, but I was just wondering if it's possible without L-Hospital's rule, derivatives, Taylor series.

Thank you in advance!

My ideas for now:

changing cosine into sine.
Maybe that. I have no other clue.

Answer

Note that

$$\lim_{y\to 0}\frac{y}{\cos\left(\frac{\pi}{2}(1+y)\right)} = \lim_{y\to 0}\frac{y}{\cos\left(\frac{\pi}{2}+\frac{\pi y}{2}\right)} = \lim_{y\to 0}\frac{y}{-\sin\left(\frac{\pi y}{2}\right)} \\ = -\frac{2}{\pi}\lim_{y\to 0}\frac{\left(\frac{\pi y}{2}\right)}{\sin\left(\frac{\pi y}{2}\right)} = -\frac{2}{\pi}(1) = -\frac{2}{\pi}$$