How to Prove the identity?

Question

$(1/cosx-tanx)^2= (1-sinx)/(1+cosx)$

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Answer 1 · 2018-06-23T08:54:07

$x=0$ is a counterexample; the correct formulation is

$(1/cos x - tan x)^2 = {1 - sin x}/{1 + sin x}$

$x=0$ is a counterexample:

$(1/cos x - tan x)^2 =(1/1 - 0)^2=1$

$(1-sinx)/(1+cos x)=(1-0)/(1+1)=1/2 quad$ NOT EQUAL

The correct formulation is

$(1/cos x - tan x)^2 = {1 - sin x}/{1 + sin x}$

Proof:

$(1/cos x - tan x)^2$

$= (1/cos x - sin x/cos x)^2$

$= ((1 - sin x)/cos x)^2$

$= (1 - sin x)^2/cos ^2 x$

$= (1 - sin x)^2/(1 - sin ^2 x)$

$= (1 - sin x)^2/{(1 - sin x)(1 + sin x)}$

$= {1 - sin x}/{1 + sin x}$