General Strategy for Simplifying Trig Identities and Expressions

Date: 12/01/2003 at 12:40:03
From: Brown
Subject: simplifying trig expressions

Simplify sin^2(x) cos^2(x) - cos^2(x).

I could not find any identities to change the problem into easier terms.

Date: 12/01/2003 at 15:09:18
From: Doctor Schwa
Subject: Re: simplifying trig expressions

Hello Brown,

Trig identities can be difficult, but you will find that as with most
things, practice pays off--the more of them you do, the better you
will get at noticing things and seeing what might work.  Keep at it!

That said, here is a somewhat standard strategy that I give students
working on trig expressions.  These are four things you can do or look
for that might help:

  1) Try converting everything to sines and cosines ... you already
have it in that form.

  2) Work with and simplify fractions, making common denominators,
dividing and so on ... no fractions here.

  3) Look for identities you recognize, like sin^2(x) + cos^2(x) = 1
... this seems to have a bit of potential in this problem, but may end
up with you going around in circles.

  4) Look for ways to factor the expression, such as common factors,
differences of squares, sums or differences of cubes and so on ...
this one will definitely help here, since there's a common factor.

Hope that strategy is helpful to you, not just on this problem, but on
many more to come!  Keep working at them--they do get easier with
practice.

- Doctor Schwa, The Math Forum
  http://mathforum.org/dr.math/