A) \[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\]
B) \[\sqrt{\frac{1-\tan x}{1+\tan x}}.{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\]
C) \[\frac{1}{2}\sqrt{\frac{1-\tan x}{1+\tan x}}.\sec \left( \frac{\pi }{4}+x \right)\]
D) None of these
Correct Answer: A
Solution :
\[y=\sqrt{\left( \frac{1+\tan x}{1-\tan x} \right)}\] or \[y=\sqrt{\tan \left( \frac{\pi }{4}+x \right)}\] \[\frac{dy}{dx}=\frac{1}{2\sqrt{\tan \left( \frac{\pi }{4}+x \right)}}{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\] \[=\frac{1}{2}\sqrt{\left[ \frac{1-\tan x}{1+\tan x} \right]}{{\sec }^{2}}\left( \frac{\pi }{4}+x \right)\].
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