Trigonometry Simplify – sin^4x – cos^4x

I thought I’d post this quick simplification using trig identities since it gave me a bit of a run for my money.  Partially so I remember it better and second, to lend a bit of help to those who might be stuck.

The Problem: Simplify sin^4 x - cos^4 x

Now, how hard could this be?!?  As it turns out, it’s not too bad, but the trick is in factoring what we have.

Factor things out: (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)

Great!  Now that we have this in better form we need to bring in a couple of trig identities which will help us solve this and one that I’m going to cite because it could trick you!

Pythagorean Identity 1: sin^2 x - cos^2 x = 1
Pythagorean Identity 2: 1 - cos^2 x = sin^2 x
Pythagorean Identity 3: sin^2 x + cos^2 x = cos(2x) (This is the tricky one!  As much as it seems like we could use this identity we would end up loosing terms.  Don’t use this)

Now, things should start looking a bit simpler to solve.  So, here we go starting with what we factored:

1) (sin^2 x - cos^2 x)(sin^2 x + cos^2 x)
2) Use Identity 1 to substitute for the first term:  (1)(sin^2 x + cos^2 x)
3) Use Identity 2 to substitute for the first part of the second term: (1)(1 - cos^2 x + cos^2 x)
4) Now combine the two like terms: (1)(1 - 2cos^2 x)
5) And so we have it!  The final result is: 1 - 2cos^2 x

Now, looking back, if you notice in identity 3, if you use it, you end up with a term (1)(cos(2x)) which multiplied out you loose the negative and the 2 in front of cos.  This is a big problem because if you did it this way, what you get in the end does not equal what you had in the beginning and thus, isn’t a valid way to simplify this problem.  As a result, you must use identities 1 and 2 as shown in the 5 steps above.

Cheers!

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