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  4. The value of sec [ tan -1( (b+a/b-a) )- tan -1( (a/b) ) ] is;

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Q. The value of $ \sec \left[ {{\tan }^{-1}}\left( \frac{b+a}{b-a} \right)-{{\tan }^{-1}}\left( \frac{a}{b} \right) \right] $ is;

KEAMKEAM 2004

Solution:

$ \sec \left[ {{\tan }^{-1}}\left( \frac{b+a}{b-a} \right)-{{\tan }^{-1}}\left( \frac{a}{b} \right) \right] $
$ =\sec \left[ {{\tan }^{-1}}\left\{ \frac{\frac{b+a}{b-a}-\frac{a}{b}}{\left( \frac{b+a}{b-a} \right)\left( \frac{a}{b} \right)} \right\} \right] $
$ =\sec [{{\tan }^{-1}}\{1\}] $
$ =\sec \frac{\pi }{4}=\sqrt{2} $