# Thread: Factor and Simplify: I'm missing a step!

1. ## Factor and Simplify: I'm missing a step!

Okay, I'm trying to simplify a trigonometric thing. I say "thing" because it's not an equation, and that has me a little confused.

Beginning: $2 sinx cosx - 4 sin^{3}x cosx$

So, I'm not given what that is equal to, but told to factor and simplify. I looked up the answer, and they gave me the last two steps.

Second-to-last: $(2 sinx cosx) (1 - 2sin^{2}x)$

Last: $(sin2x) (cos2x)$

Okay, getting from the second-to-last step to the last is easy, using the double-angle formulae. I just don't know how to get from the beginning to the second-to-last step! I checked it out in MS Excel, and

$2 sinx cosx - 4sin^{3}x cosx$ does in fact equal $(2 sinx cosx) (1 - 2sin^{2}x)$

... I just don't know how they got there. Is it that there is some common factor in the first two terms of the original thing? Can I divide each side of a subtraction operation in half or something?

2. $2sinxcosx - 4sin^{3}xcosx$

You can factor out 2sinxcosx from that entire expression. Multiply it back in and maybe you'll get what they're doing.:
$\underbrace{2\sin x \cos x}_{\sin 2x} \underbrace{(1 - 2\sin^{2}x)}_{\cos2x}$

Hopefully you'll recognize the expressions for sin2x and cos2x.

3. 2 sinx cosx - 4 sin^{3}x cosx =
2 sinx cosx - 2 sinx cosx (2 sin^{2}x) =
2 sinx cosx (1 - 2 sin^{2}x)

4. Oh thanks! Hop David has supplied the steps I was missing.

Let me reiterate to see if I have it right. First, I can pull out (2 sin^{2}x) from 4 sin^{3}x cosx. That leaves me with 2 sinx cosx on both sides of my subtraction sign. I missed that at first.

Next, the point I REALLY missed, was that A - AB = A(1-B). It would be so easy to go the other way, but my reverse gears are rusty. Thus I can go from
2 sinx cosx - 2 sinx cosx (2 sin^{2}x)
to
2 sinx cosx (1 - 2 sin^{2}x).

THEN I can use my double-angle identities. Geez, I've done like twenty problems in my book after that one ... they're all cakewalks by comparison.