## Small trig substitution problem.

1. The problem statement, all variables and given/known data

I was working on a problem set involving greens theorem and I came across this peculiar trig substitution. I was just wondering how it came about as I couldn't find anything like it on Wikipedia's page.

$sin^4(t)cos^2(t) + cos^4(t) sin^2(t) = cos^2(t)sin^2(t)$

3. The attempt at a solution
I tried using the basic's such as $(cos^2(t))^2 = (1 - sin^2(t))^2$

along with $(sin^2(t))^2 = (1 - cos^2(t))^2$

which after some substitution gives

$cos^6(t) - cos^4(t) + sin^2(t)cos^2(t) + sin^6(t) - sin^4(t) + sin^2(t)cos^2(t)$

Which is close to what I wanted, but I started to get the feeling that the path I was going down wasn't going to yield my identity. Can anyone shed some light?
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 Hi ozone! Did you try the simpler idea of taking $$sin^2t\cdot cos^2t$$ out common? Edit : Arrgh! multi-post Mod note: not any more...

Mentor
 Quote by ozone 1. The problem statement, all variables and given/known data I was working on a problem set involving greens theorem and I came across this peculiar trig substitution. I was just wondering how it came about as I couldn't find anything like it on Wikipedia's page. $sin^4(t)cos^2(t) + cos^4(t) sin^2(t) = cos^2(t)sin^2(t)$ 3. The attempt at a solution I tried using the basic's such as $(cos^2(t))^2 = (1 - sin^2(t))^2$ along with $(sin^2(t))^2 = (1 - cos^2(t))^2$ which after some substitution gives $cos^6(t) - cos^4(t) + sin^2(t)cos^2(t) + sin^6(t) - sin^4(t) + sin^2(t)cos^2(t)$ Which is close to what I wanted, but I started to get the feeling that the path I was going down wasn't going to yield my identity. Can anyone shed some light?

$\sin^4(t)\cos^2(t) + \cos^4(t) \sin^2(t) = \cos^2(t)\sin^2(t)\left(\sin^2(t)+\cos^2(t)\right) \ ?$

## Small trig substitution problem.

Thanks sammy's that is definitely sufficient proof for me. DOH that was an easy one =d

edit: thanks infinitum too you would have pointed me in the right direction

 Quote by ozone Thanks sammy's that is definitely sufficient proof for me. DOH that was an easy one =d edit: thanks infinitum too you would have pointed me in the right direction
Err, what SammyS and I said are exactly the same thing. I preferred not to elaborate

 Quote by Infinitum Hi ozone! Did you try the simpler idea of taking $$sin^2t\cdot cos^2t$$ out common? Edit : Arrgh! multi-post
I second that motion. Always factor factor FACTOR !!

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