How do you determine if #secx -cscx# is an even or odd function?

1 Answer
George C.
Sep 11, 2016

#sec x - csc x# is neither even nor odd.

Explanation:

  • An even function is one for which #f(-x) = f(x)# for all #x# in its domain.
  • An odd function is one for which #f(-x) = -f(x)# for all #x# in its domain.

Consider #f(x) = sec(x) - csc(x)# with #x=pi/4# ...

Note that:

#sec(pi/4) = 1/(cos(pi/4)) = 1/(1/(sqrt(2))) = sqrt(2)#

#csc(pi/4) = 1/(sin(pi/4)) = 1/(1/(sqrt(2))) = sqrt(2)#

#sec(-pi/4) = 1/(cos(-pi/4)) = sqrt(2)#

#csc(-pi/4) = 1/(sin(-pi/4)) = -sqrt(2)#

#f(-x) = sec(-pi/4) - csc(-pi/4) = sqrt(2)-(-sqrt(2)) = 2sqrt(2)#

#f(x) = sec(pi/4) - csc(pi/4) = sqrt(2) - sqrt(2) = 0#

So neither #f(-x) = f(x)# nor #f(-x) = -f(x)#

So #f(x)# is neither even nor odd.

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