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)

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

$R H S = \frac{1 - \sin \left(x\right)}{1 + \sin \left(x\right)}$

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

$= \frac{\left(1 - \sin \left(x\right)\right) \left(1 - \sin \left(x\right)\right)}{1 - {\sin}^{2} \left(x\right)}$

$= \frac{\left(1 - \sin \left(x\right)\right) \left(1 - \sin \left(x\right)\right)}{{\cos}^{2} \left(x\right)}$

$= \frac{1 + {\sin}^{2} \left(x\right) - 2 \sin \left(x\right)}{{\cos}^{2} \left(x\right)}$

$= \frac{1}{\cos} ^ 2 \left(x\right) + {\sin}^{2} \frac{x}{\cos} ^ 2 \left(x\right) - 2 \sin \frac{x}{\cos} ^ 2 \left(x\right)$

$= {\sec}^{2} \left(x\right) + {\tan}^{2} \left(x\right) - 2 \tan \left(x\right) \sec \left(x\right)$

$= {\left(\sec \left(x\right) - \tan \left(x\right)\right)}^{2} = L H S$

The two sides are equal, hence an identity