Question:
If $\frac{\pi}{2} (a) 2 sec x (b) −2 sec x (c) sec x (d) −sec x
Solution:
(b) −2 sec x
$\sqrt{\frac{1-\sin x}{1+\sin x}}+\sqrt{\frac{1+\sin x}{1-\sin x}}$
$=\sqrt{\frac{(1-\sin x)(1-\sin x)}{(1+\sin x)(1-\sin x)}}+\sqrt{\frac{(1+\sin x)(1+\sin x)}{(1-\sin x)(1+\sin x)}}$
$=\sqrt{\frac{(1-\sin x)^{2}}{1-\sin ^{2} x}}+\sqrt{\frac{(1+\sin x)^{2}}{1-\sin ^{2} x}}$
$=\sqrt{\frac{(1-\sin x)^{2}}{\cos ^{2} x}}+\sqrt{\frac{(1+\sin x)^{2}}{\cos ^{2} x}}$
$=\frac{(1-\sin x)}{-\cos x}+\frac{(1+\sin x)}{-\cos s x} \quad\left[\frac{\pi}{2}
$=-(\sec x-\tan x)-(\sec x+\tan x)$
$=-2 \sec x$
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