$ \displaystyle{\lim_{x \to (-1)^-}} \frac{\cos \left( \frac{\pi }{2}x \right)}{1+\sin \left( \frac{\pi }{2}x \right)} \displaystyle{\lim_{x \to (-1)^-}}- \frac{\cos \left( \frac{\pi }{2}x \right)}{1+\sin \left( \frac{\pi }{2}x \right)}\times \frac{1-\sin \left( \frac{\pi }{2}x \right)}{1-\sin \left( \frac{\pi }{2}x \right)} $ $= \displaystyle{\lim_{x \to (-1)^-}} \frac{\cos \left( \frac{\pi }{2}x \right)\left( 1-\sin \left( \frac{\pi }{2}x \right) \right)}{{{\cos }^{2}}\left( \frac{\pi }{2}x \right)}=\displaystyle{\lim_{x \to (-1)^-}} \frac{1-\sin \left( \frac{\pi }{2}x \right)}{\cos \left( \frac{\pi }{2}x \right)}\displaystyle{\lim_{x \to (-1)^-}} =\frac{{{2}^{-}}}{{{0}^{-}}}=-\infty $