687. $f(x)=\ln \sqrt{\frac{1-\sin \alpha}{1+\sin \alpha}} ;$ calculate $f^{\prime}\left(\frac{\pi}{4}\right)$.
Solution. Here it is appropriate to first take the logarithm. After logarithmization, we get

$$
f(x)=\frac{1}{2}[\ln (1-\sin \alpha)-\ln (1+\sin \alpha)]
$$

Now we compute the derivative

$$
f^{\prime}(x)=\frac{1}{2}\left[\frac{-\cos \alpha}{1-\sin \alpha}-\frac{\cos \alpha}{1+\sin \alpha}\right]=-\frac{1}{\cos \alpha}
$$

By setting \( a=\frac{\pi}{4} \), we find \( f^{\prime}\left(\frac{\pi}{4}\right)=-\frac{1}{\cos \frac{\pi}{4}}=-\sqrt{2} \).