How do you simplify #(1+cos x) (1-cos x)#?
Expand the brackets using FOIL , or the method you use.
#rArr (1 + cosx)(1 - cosx) = 1 -cosx + cosx - cos^2 x #
#= 1 - cos^2 x#
using the identity
#color(red)(|bar(ul(color(white)(a/a)color(black)( sin^2 x + cos^2 x = 1 )color(white)(a/a)|)))#
# 1 - cos^2 x = sin^2 x#
The expression we have fits the difference of squares pattern
We know that a difference of squares pattern is equal to
This expression should look familiar. It is derived from the Pythagorean Identity
where we can subtract
Thus, this expression is equal to
All we did was use the difference of squares property to our advantage, recognize that the expression we had is derived from the Pythagorean Identity, use it, and simplify.
Hope this helps!