Small trig substitution problem.

[ozone]

Homework Statement

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.

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

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?

 

[Infinitum]

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...

 

[SammyS]

Homework Statement

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.

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

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?

How about:

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

 

[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

 

[Infinitum]

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

 

[Skins]

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 !