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
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