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If u = {\cot ^{ - 1}}\sqrt {\tan \alpha } - {\tan ^{ - 1}}\sqrt {\tan \alpha }, then \tan \;\left( {\frac{\pi }{4} - \frac{u}{2}} \right) is equal to

Option: 1

\sqrt {\tan \alpha }


Option: 2

\sqrt {\cot \alpha }


Option: 3

tan a


Option: 4

cot a


Answers (1)

best_answer

As we learnt

 

Important Results of Inverse Trigonometric Functions -

 

\cot ^{-1}\left ( \cot \Theta \right )= \Theta

- wherein

if 0< \Theta < \pi

 

 

Let \sqrt {\tan \alpha } = \tan x\;,\;  then u = cot^{-1} (tanx) - tan^{-1} (tanx)

=  \frac{\pi }{2} - x - x = \frac{\pi }{2} - 2x

\Rightarrow         = 2x = \frac{\pi }{2} - u   \Rightarrow x = \frac{\pi }{4} - \frac{u}{2}

\Rightarrow         \tan x = \tan \,\left( {\frac{\pi }{4} - \frac{u}{2}} \right)

\Rightarrow         \sqrt {\tan \alpha } = \tan \;\left( {\frac{\pi }{4} - \frac{u}{2}} \right)

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