MCQ
Let $0 < x < \frac{\pi }{4}.$ Then $\sec 2x - \tan 2x = $
  • A
    $\tan \left( {x - \frac{\pi }{4}} \right)$
  • $\tan \left( {\frac{\pi }{4} - x} \right)$
  • C
    $\tan \left( {x + \frac{\pi }{4}} \right)$
  • D
    ${\tan ^2}\left( {x + \frac{\pi }{4}} \right)$

Answer

Correct option: B.
$\tan \left( {\frac{\pi }{4} - x} \right)$
b
(b) $\sec 2x - \tan 2x = \frac{{1 - \sin 2x}}{{\cos 2x}}$ 

$ = \frac{{{{(\cos x - \sin x)}^2}}}{{({{\cos }^2}x - {{\sin }^2}x)}} $

$= \frac{{\cos x - \sin x}}{{\cos x + \sin x}} = \frac{{1 - \tan x}}{{1 + \tan x}}$

$ = \frac{{\tan \frac{\pi }{4} - \tan x}}{{1 + \tan \left( {\frac{\pi }{4}} \right)\sin x}} = \tan \left( {\frac{\pi }{4} - x} \right)$.

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