Common Mistake in Simplifying Trig Expressions

Date: 12/07/2003 at 19:46:12
From: Stacy 
Subject: How do I simplify the trigonometric expression

I don't know how to simplify this problem:

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

I think the sines^2 cancel so then I am left with -cos^2(X) / -sin(x)
cos(x) so then you will have cos(x) / sin(x) and that equals to
cot(x).  But the answer in the book is 1 + cot(x).

Date: 12/08/2003 at 10:31:49
From: Doctor Riz
Subject: Re: How do I simplify the trigonometric expression

Hi Stacy -

Thanks for writing Dr. Math.  You've done a nice job of sharing your 
thinking, which is great!  Unfortunately, you are making a mistake
which many people make in this situation.  Let's take a look. 

I don't agree that you can cancel out the sines^2 as you say.  The
reason you can't is that you have subtractions in the numerator and
denominator.  You can only cancel stuff like that when the entire
numerator and entire denominator are each made up of pieces being
multiplied together.  Let's look at some examples with numbers to show
what I mean:

Suppose you have this fraction:

  8 - 5
  -----
  8 - 2

We can clearly see that the numerator is worth 3 (8 - 5) and the 
denominator is worth 6 (8 - 2).  Thus:

  8 - 5    3     1
  ----- = --- = ---
  8 - 2    6     2

Now, what you are doing with your sines^2 would be like looking at
this problem and saying that you can cancel out the 8's.  But if you
cancel the 8's you are left with

  - 5
  ----, which can't be right since we know it should be 1/2
  - 2

As a comparison, suppose the numbers are being multiplied rather than 
subtracted.  Now the problem becomes:

  8 * 5   40    5
  ----- = -- = ---  (when reduced)
  8 * 2   16    2

If we cancel out the 8's in this case, we are left with:

  8 * 5   1 * 5    5
  ----- = ----- = ---
  8 * 2   1 * 2    2

It's the same answer, so this reducing or canceling is fine.  This is
a very common mistake, so don't feel bad.  The thing to keep in mind
is that if you have more than one term in your numerator or
denominator, you can't just cancel one of those terms out.  Before
reducing fractions, you want to make sure the numerator and
denominator are both made up of pieces (factors) being multiplied.

So let's think about your problem.  If we use some factoring, we can 
turn both the numerator and denominator into multiplications, which 
will allow us to do some canceling or reducing.

Notice that sin^2(x) - cos^2(x) can be thought of as a difference of 
two squares, and factored accordingly:

  a^2 - b^2 = (a + b)(a - b), so 

  sin^2(x) - cos^2(x) = (sinx + cosx)(sinx - cosx)

Your denominator can be factored by factoring out a sinx:

  sin^2(x) - sinxcosx = (sinx)(sinx - cosx)

So notice that we have now changed a problem that had subtractions in 
the numerator and denominator into one with multiplications in both 
those places:

  sin^2(x) - cos^2(x)   (sinx + cosx)(sinx - cosx)
  ------------------- = --------------------------
  sin^2(x) - sinxcosx      (sinx)(sinx - cosx)

At this point, if there are any two factors in (  ) that match in the 
numerator and denominator, they can be cancelled out.  Do you see 
anything that will cancel?

Again, the idea is to factor the numerator and denominator completely
to turn any addition/subtraction into multiplication.  That doesn't
mean that within a particular factor there won't be addition or 
subtraction--in fact, 3 of the 4 factors involved in this problem 
contain addition or subtraction.  But by looking at the whole factors 
as individual pieces, we have two things being multiplied in the 
numerator and two in the denominator:

  (  )(  )
  --------
  (  )(  )

We can cancel any pair of (  ) that are identical if one is in the 
numerator and one is in the denominator.  We can NOT go inside one of
those (  ) and cancel just part of it out--it has to be the whole 
(  ) or nothing.

So once you've canceled out, you have two more steps to get to the 
answer in the book.  Can you see how to do it?

Hope this helps.  Good luck, and write back if you still need more
help on this idea.

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