Question

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1-\sin x}{{\cos }^{2}x}$

Answer 1

$$\begin{gathered} \underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1-\sin x}{{\cos }^{2}x} \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\left(1-\sin x\right)}{\left(1-{\sin }^{2}x\right)} \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\left(1-\sin x\right)}{\left(1-\sin x\right)\left(1+\sin x\right)} \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\left(\frac{1}{1+\sin x}\right) \\ \Rightarrow \frac{1}{2} \end{gathered}$$