Question

$\underset{x\to \pi /4}{lim}\frac{\tan x-1}{\cos 2x}$ is equal to

• A. $1$
• B. $0$
• C. $-2$
• D. $-1$

Answer 1

The correct option is D $-1$

$\underset{x\to \pi /4}{lim}\frac{\tan x-1}{\cos 2x}=\underset{h\to 0}{lim}\frac{\tan (\frac{\pi }{4}+h)-1}{\cos 2(\frac{\pi }{4}+h)}$
$[\because x=\frac{\pi }{4}+h]$

$=\underset{h\to 0}{lim}\frac{\left(\frac{1+\tan h}{1-\tan h}\right)-1}{\cos (\frac{\pi }{2}+2h)}$

$=\underset{h\to 0}{lim}\frac{1+\tan h-1+\tan h}{-\sin 2h(1-\tan h)}$
$=\underset{h\to 0}{lim}\frac{-2\tan h}{2\sin h\cos h(1-\tan h)}$
$=\underset{h\to 0}{lim}\frac{-1}{{\cos }^{2}h(1-\tan h)}=-1$

Hence, option D is correct.