How do you verify the identity $(1 + sin α) (1 - sin α) = {cos}^{2} α$ $(1+sinalpha)(1-sinalpha)=cos^2alpha$ ?

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Answer 1 · 2017-01-04T21:11:34

see below

$(1 + sin α) (1 - sin α) = {cos}^{2} α$ $color(red)((1+sinalpha)(1-sinalpha))=cos^2alpha$

apply FOIL to the red bit. you get: $1 - sin α + sin α - {sin}^{2} α = 1 - {sin}^{2} α$ $1 - sin alpha + sin alpha - sin^2 alpha = 1 - sin^2 alpha$

then you have the Pythagorean identity to finish it off :)

$1 - {sin}^{2} α = {cos}^{2} α \Rightarrow {sin}^{2} α + {cos}^{2} α = 1$ $1 - sin^2 alpha = cos^2 alpha implies sin^2 alpha + cos^2 alpha = 1$

How do you verify the identity (1+sinalpha)(1-sinalpha)=cos^2alpha(1+sinα)(1−sinα)=cos2α(1+sinalpha)(1-sinalpha)=cos^2alpha?