Question

(1+cos x +sin x)/(1+cos x-sin x)=( 1+ sin x)/cos x

Answer 1

To prove :

(1+cos x+sin x) / (1+cos x-sin x) = (1+sin x)/cos x

Proof :

Taking LHS

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LHS = (1+cos x+sin x) / (1+cos x-sin x )

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dividing and multiplying by cos x

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LHS = (1+cos x+sin x) / (1+cos x-sin x) × (cos x)/(cos x)

LHS = (sec x + 1 + tan x)/(sec x + 1 - tan x)

LHS = (sec x + tan x + 1)/(sec x - tan x + 1)

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using trigonometric identity

sec²A - tan²A = 1

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LHS = (sec x + tan x + (sec²x - tan²x))/(sec x - tan x + 1)

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using algebraic identity

( a² - b² ) = ( a + b ) ( a - b )

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LHS = (sec x + tan x + ( sec + tan x ) (sec x - tan x ) / (sec x - tan x + 1)

LHS = ( sec x + tan x ( 1 + sec x - tan x )) / (sec x - tan x + 1)

LHS = ( sec x + tan x ( 1 + sec x - tan x ))/(1 + sec x - tan x)

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(1 + sec x - tan x) will be divided by (1 + sec x - tan x)

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LHS = sec x + tan x

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putting sec x = 1/cos x , tan x = sin x / cos x

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LHS = (1/cos x) + (sin x / cos x)

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Taking LCM

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LHS = ( 1 + sin x ) / cos x = RHS

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Hence, PROVED.

Answer 2

Answer:

★ see this attachment .