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

If \sec x + \tan x = 2012

then 2011 (cosec x + \cot x) is equal to

(A) 2011

(B) 2012

(C) 2013

(D) \frac{2011}{2013}

(E) \frac{2013}{2012}

Answered by abwayh last updated on 29/Sep/17

it just try;

\sec x + \tan x = 2012

(\sec x - 1) + \tan x = 2011; \sec x + \tan x + 1 = 2013

2 \tan x = 2011 \dots \dots (i); 2 \sec x = 2013 \dots . . (ii)

2011 (cosec x + \cot x) =

= 2 \tan x (\frac{1}{\sin x} + \frac{\cos x}{\sin x})

= 2 (\frac{1}{\cos x} + 1)

= 2 \sec x + 2

= 2013 + 2

= 2015

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

I f u w o n t e x p l a i n t h i s I w o u l d

n e v e r u n d e r s t a n d h o w (i)

& (i i) a r e f o r m e d .

Commented by Tinkutara last updated on 29/Sep/17

A n o t h e r w a y :

S i n c e \sec^{2} x - \tan^{2} x = 1

\Rightarrow (\sec x - \tan x) (\sec x + \tan x) = 1

S o i f \sec x + \tan x = 2012, t h e n

\sec x - \tan x = \frac{1}{2012}

A d d i n g a n d s u b t r a c t i n g w e c a n g e t

\sec x a n d \tan x a n d t h u s a l l

t r i g o n o m e t r i c r a t i o s .

Commented by Joel577 last updated on 29/Sep/17

\sec x - 1 \neq \tan x

\tan x + 1 \neq \sec x

Commented by Joel577 last updated on 30/Sep/17

T h a n k s f o r t h e i d e a

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 (cosec x + \cot x) = 2013

Commented by Joel577 last updated on 30/Sep/17

T h a n k y o u v e r y m u c h . Y o u m a d e i t v e r y s i m p l e

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