An even function is a function satisfying:
#f(-x) = f(x)# for any #x#
An odd function is a function satisfying:
#f(-x) = -f(x)# for any #x#
Note that #cos(x)# is even since #cos(-x) = cos(x)# for any #x#
Note that #sin(x)# is odd since #sin(-x) = -sin(x)# for any #x#
Let #f(x) = cos(x) sin(x)#
Then for any #x#:
#f(-x)#
#= cos(-x) * sin(-x)#
#= cos(x) * (-sin(x))#
#= - cos(x)sin(x)#
#= -f(x)#
So #f(x) = cos(x) sin(x)# is an odd function.
In general, if #e(x)# is an even function and #o(x)# is an odd function, then #e(x)o(x)# is odd too. The odd/even result you get when multiplying functions is like the result when you add even and odd numbers.
The functions #e(o(x))#, #o(e(x))# and #e(e(x))# will all be even, but the function #o(o(x))# will be odd. The odd/even result you get when composing functions is like the result when you multiply even and odd numbers.
