How do you verify the identify (1-sintheta)(1+sintheta)=cos^2theta?

Aug 27, 2016

We will use the following well known identities:

• ${\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right) = 1$
• ${a}^{2} - {b}^{2} = \left(a + b\right) \left(a - b\right)$

With those, starting from the left hand side:

$\left(1 - \sin \left(\theta\right)\right) \left(1 + \sin \left(\theta\right)\right)$

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

$= \left({\sin}^{2} \left(\theta\right) + {\cos}^{2} \left(\theta\right)\right) - {\sin}^{2} \left(\theta\right)$

$= {\cos}^{2} \left(\theta\right)$