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Question Number 123375 by bemath last updated on 25/Nov/20

$$\underset{x\to \pi /2}{\lim }(\sec {}^{2}x-\sec x\tan x)=?$$

Answered by Dwaipayan Shikari last updated on 25/Nov/20

$$\underset{x\to \frac{\pi }{2}}{\lim }\left(\frac{1-sinx}{{cos}^{2}x}\right)=\frac{2\left({sin}^2(\frac{\pi }{4}-\frac{x}{2})\right)}{{sin}^2(\frac{\pi }{2}-x)}=\frac{2{(\frac{\pi }{4}-\frac{x}{2})}^2}{{(\frac{\pi }{2}-x)}^2}=2.\frac{1}{4}=\frac{1}{2}$$

Answered by liberty last updated on 25/Nov/20

$$\underset{x\to \pi /2}{\lim sec}x(\sec x-\tan x)=$$$$\underset{x\to \pi /2}{\lim }\frac{\left(\frac{1-\sin x}{\cos x}\right)}{\cos x}=\underset{x\to \pi /2}{\lim }\frac{1-\sin x}{\cos {}^{2}x}$$$$[letx=\frac{\pi }{2}+y;y\to 0]$$$$\underset{y\to 0}{\lim }\frac{1-\cos y}{\sin {}^{2}y}=\underset{y\to 0}{\lim }\frac{2\sin {}^2\left(y/2\right)}{\sin {}^{2}y}=\frac{1}{2}$$

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