establish the identity of (1-sec x / tan x) + (tan x / 1- sec x ) =-2 csc x

Question

establish the identity of (1-sec x  / tan x) + (tan x / 1- sec x ) =-2 csc x

ybarrap · Answer

$$(1-\sec x)  / \tan x + \tan x / (1- \sec x ) $$
$$u = \tan x/(1-\sec x)$$
Equation has form $$1/u + u$$
$$=(u^2 + 1)/u$$
$$\frac{ (\tan x/ (1-\sec x)) ^2+1 }{\tan x/ (1-\sec x)  }$$
$$=\frac{ \sin^2 x/(cosx -1)^2 + 1 }{ \tan x/ (1-\sec x) }$$
$$=\frac{ \sin^2 x + \cos^2 x - 2\cos x +1 }{ \tan x/ (1-\sec x)  }$$
$$=\frac{ 2-2cosx }{ sinx/(\cos x(1-1/\cos x) }$$
$$=\frac{ \csc x(2 -2\cos x) }{ \cos x -1 }$$
$$=\frac{ 2\csc x (1-\cos x) }{ -(1-\cos x) }$$
$$=-2\csc x$$

ybarrap · Answer

i left a few steps out for you to fill in

anonymous · Answer

yeah solving with the u is a lot easier thanks so much

ybarrap · Answer

np