Question #d3713

1 Answer
salamat
Jun 7, 2017

see explanation

Explanation:

Q : # (sin x + 1 )/ cos x = (((tan x + sec x) -1) / ((tan x + 1) - sec x)) * (((tan x + sec x) + 1) / ((tan x + 1) + sec x))#

Let we take RHS to prove LHS,

#(((tan x + sec x) -1) / ((tan x + 1) - sec x)) * (((tan x + sec x) + 1) / ((tan x + 1) + sec x)) = ((tan x + sec x)^2 -1) / ((tan x + 1)^2 - sec^2 x) #

#=(tan^2 x + 2tan x sec x + sec ^2 x -1) / (tan^2 x + 2tan x + 1 - sec^2 x #

rearrange,
#=(tan^2 x + 2tan x sec x +( sec ^2 x -1)) / (tan^2 x + 1 + 2tan x - (sec^2 x) #

replace #sec ^2 x -1 = tan^2 x, sec ^2 x = tan^2 x + 1# in the equation
#=(tan^2 x + 2tan x sec x + tan^2 x) / (tan^2 x + 1 + 2tan x - (tan^2 x + 1) #

#=(2tan^2 x + 2tan x sec x ) / ( 2tan x ) #

#= tan x + sec x = sin x/ cos x + 1/cos x #

#= (sin x + 1)/cos x# #-># proved

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