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}#