Question

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

Answer 1

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

Answer 2

i left a few steps out for you to fill in

Answer 3

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

Answer 4

np