Evaluate $ \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{\cos(y)-\sin(2y)}{\cos(x)\cos(y)}$
So far my steps have included: $ = \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{\cos(y)-2\sin(y)\cos(y)}{\cos(x)\cos(y)}$ using the double angle identity
$ = \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{\cos(y)(1-2\sin(y))}{\cos(x)\cos(y)} = \lim\limits_{(x,y)\rightarrow(\pi,\frac{\pi}{2})} \frac{1-2\sin(y)}{\cos(x)}$, which allows us to take the limit at the point since the point exists within the domain, so:
$ = \frac{1-2\sin(\frac{\pi}{2})}{\cos(\pi)} = \frac{-1}{-1} = 1$
There is no solution included in my textbook so I was hoping someone might verify. I was also wondering if this solution was sufficient or if there were further steps to evaluate the limit, since other questions require taking different paths to confirm the limit existed from every direction. Would this be required here? Thank you in advance!