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Question Number 21622 by Joel577 last updated on 29/Sep/17

$$\mathrm{If}\sec x+\tan x=2012$$$$\mathrm{then}2011(\mathrm{cosec}x+\cot x)\mathrm{is}\mathrm{equal}\mathrm{to}$$$$\left(A\right)2011$$$$\left(B\right)2012$$$$\left(C\right)2013$$$$\left(D\right)\frac{2011}{2013}$$$$\left(E\right)\frac{2013}{2012}$$

Answered by abwayh last updated on 29/Sep/17

$$\mathrm{it}\mathrm{just}\mathrm{try};$$$$\sec \mathrm{x}+\tan \mathrm{x}=2012$$$$(\sec \mathrm{x}-1)+\tan \mathrm{x}=2011;\sec \mathrm{x}+\tan \mathrm{x}+1=2013$$$$2\tan \mathrm{x}=2011......\left(\mathrm{i}\right);2\sec \mathrm{x}=2013.....\left(\mathrm{ii}\right)$$$$2011(\mathrm{cosec}\mathrm{x}+\cot \mathrm{x})=$$$$=2\tan \mathrm{x}(\frac{1}{\sin \mathrm{x}}+\frac{\cos \mathrm{x}}{\sin \mathrm{x}})$$$$=2(\frac{1}{\cos \mathrm{x}}+1)$$$$=2\sec \mathrm{x}+2$$$$=2013+2$$$$=2015$$$$$$

Commented by $@ty@m last updated on 29/Sep/17

$$IfuwontexplainthisIwould$$$$neverunderstandhow\left(i\right)$$$$\mathrm{\&}\left(ii\right)areformed.$$

Commented by Tinkutara last updated on 29/Sep/17

$$Anotherway:$$$$Since{\sec }^{2}x-{\tan }^{2}x=1$$$$\Rightarrow (\sec x-\tan x)(\sec x+\tan x)=1$$$$Soif\sec x+\tan x=2012,then$$$$\sec x-\tan x=\frac{1}{2012}$$$$Addingandsubtractingwecanget$$$$\sec xand\tan xandthusall$$$$trigonometricratios.$$

Commented by Joel577 last updated on 29/Sep/17

$$\sec x-1\ne \tan x$$$$\tan x+1\ne \sec x$$

Commented by Joel577 last updated on 30/Sep/17

$$Thanksfortheidea$$

Answered by $@ty@m last updated on 29/Sep/17

$$\sec x+\tan x=2012$$$$\frac{1}{\cos x}+\frac{\sin x}{\cos x}=2012$$$$\frac{1+\sin x}{\cos x}=2012$$$$\frac{{(\cos \frac{x}{2}+\sin \frac{x}{2})}^2}{{\cos }^2\frac{x}{2}-{\sin }^2\frac{x}{2}}=2012$$$$\frac{\cos \frac{x}{2}+\sin \frac{x}{2}}{\cos \frac{x}{2}-\sin \frac{x}{2}}=2012$$$$\frac{1+\tan \frac{x}{2}}{1-\tan \frac{x}{2}}=2012$$$$1+\tan \frac{x}{2}=2012(1-\tan \frac{x}{2})$$$$2013\tan \frac{x}{2}=2011$$$$2011\cot \frac{x}{2}=2013$$$$2011\frac{\cos \frac{x}{2}}{\sin \frac{x}{2}}=2013$$$$2011\frac{2\cos \frac{x}{2}\cos \frac{x}{2}}{2\sin \frac{x}{2}\cos \frac{x}{2}}=2013$$$$2011\frac{{2\cos }^2\frac{x}{2}}{\sin x}=2013$$$$2011\frac{1+\cos x}{\sin x}=2013$$$$2011(\mathrm{cosec}x+\cot x)=2013$$$$$$$$$$

Commented by Joel577 last updated on 30/Sep/17

$$Thankyouverymuch.Youmadeitverysimple$$

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