4. Equation or Identity? Is the following relationship an equation or an identity? If an equation, solve it. If an identity, prove it. #(secx-tanx)^2 = (1-sinx)/(1+sinx)#

1 Answer
Øko
Mar 6, 2018

Identity

Explanation:

Usually i start by plugging in some nice values,
if it is true for these values try proving it as an identity,
in this example it is true for #x=0# and #x=pi#

So we try proving it as an identity

#RHS=(1-sin(x))/(1+sin(x))#

#=((1-sin(x))(1-sin(x)))/((1+sin(x)(1-sin(x))#

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

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

#=(1+sin^2(x)-2sin(x))/(cos^2(x))#

#=1/cos^2(x)+sin^2(x)/cos^2(x)-2sin(x)/cos^2(x)#

#=sec^2(x)+tan^2(x)-2tan(x)sec(x)#

#=(sec(x)-tan(x))^2=LHS#

The two sides are equal, hence an identity

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