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

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)