Prove that \[ \sqrt{\frac{1-\sin x}{1+\sin x}} + \sqrt{\frac{1+\sin x}{1-\sin x}} = -\frac{2}{\cos x} \] where \[ \frac{\pi}{2}

Solution:

\[ \sqrt{\frac{1-\sin x}{1+\sin x}} + \sqrt{\frac{1+\sin x}{1-\sin x}} \]

Taking LCM,

\[ = \frac{ (1-\sin x)+(1+\sin x) } { \sqrt{(1-\sin x)(1+\sin x)} } \]

\[ = \frac{2} {\sqrt{1-\sin^2 x}} \]

\[ = \frac{2}{\sqrt{\cos^2 x}} \]

\[ = \frac{2}{|\cos x|} \]

Since \[ \frac{\pi}{2}

Therefore,

\[ |\cos x|=-\cos x \]

\[ \frac{2}{|\cos x|} = -\frac{2}{\cos x} \]

Hence proved.