Prove the identity: (tanx+1)/(tanx) - (secxcscx+1)/(tanx+1) = (cos x)/(sinx +cosx)

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justaguide | College Teacher | (Level 2) Distinguished Educator

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The identity that has to be proved is : (tan x + 1)/(tan x) - (sec x * csc x + 1)/(tan x + 1) = (cos x)/(sin x + cos x).

We know that tan x = sin x / cos x , sec x = 1/cos x and csc x = 1/sin x.

Let's start from the left hand side:

(tan x + 1)/(tan x) - (sec x * csc x + 1)/(tan x + 1)

=> (tan x/tan x + 1/tan x) - (sec x * csc x + 1)/((sin x/cos x)+ 1)

=> (tan x/tan x + 1/tan x) - (sec x * csc x + 1)(cos x)/(sin x + cos x)

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

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

=> ((sin x + cos x)^2 - (1 + cos x * sin x))/(sin x)(sin x + cos x)

=> ((sin x)^2 + (cos x)^2 + 2*sin x * cos x - 1 - cos x * sin x)/(sin x)(sin x + cos x)

Use (sin x)^2 + (cos x)^2 = 1

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

=> (sin x * cos x)/(sin x)(sin x + cos x)

=> (cos x)/(sin x + cos x)

which is the right hand side.

This proves that (tan x + 1)/(tan x) - (sec x * csc x + 1)/(tan x + 1) = (cos x)/(sin x + cos x)

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