# Thread: Proving a Couple of Trig Identities

1. ## Proving a Couple of Trig Identities

sin^2x - tan^2x = -sin^2x tan^2x

I kept working on tan on both sides, but cannot seem to get them to equal.

also this one

sin^4x + 2sin^2x.cos^2x + cos^4x = 1

I went :
sin^2x.sin^2x + 2sin^2x.cos^2x + cos^2x.cos^2x

I tried to make middle one sin2x didn't seem to get me anywhere.
Also tried changing one of the left sin^2x into 1-cos^2x and
one of the right cos^2x into 1-sin^2x

but they didn't cancel each other out.

2. ## re: Proving a Couple of Trig Identities

Originally Posted by Dante
sin^2x - tan^2x = -sin^2x tan^2x

I kept working on tan on both sides, but cannot seem to get them to equal.

also this one

sin^4x + 2sin^2x.cos^2x + cos^4x = 1

I went :
sin^2x.sin^2x + 2sin^2x.cos^2x + cos^2x.cos^2x

I tried to make middle one sin2x didn't seem to get me anywhere.
Also tried changing one of the left sin^2x into 1-cos^2x and
one of the right cos^2x into 1-sin^2x

but they didn't cancel each other out.
$\displaystyle \sin^2{x} - \tan^2{x} =$

$\displaystyle \sin^2{x}\left(1 - \frac{1}{\cos^2{x}}\right) =$

$\displaystyle \sin^2{x}\left(1 - \sec^2{x}\right) =$

$\displaystyle \sin^2{x}\left(-\tan^2{x}\right) = -\sin^2{x}\tan^2{x}$

$\displaystyle \sin^4{x} +2\sin^2{x}\cos^2{x} + \cos^4{x} =$

$\displaystyle \left(\sin^2{x} + \cos^2{x}\right)^2 = \, ?$

3. ## re: Proving a Couple of Trig Identities

Originally Posted by Dante
[B]sin^2x - tan^2x = -sin^2x tan^2x[/B
$\displaystyle \sin^2(x)-\tan^2(x)=\frac{\sin^2(x)(\cos^2(x)-1)}{\cos^2(x)}$

4. ## re: Proving a Couple of Trig Identities

Originally Posted by skeeter
$\displaystyle \sin^2{x} - \tan^2{x} =$

$\displaystyle \sin^2{x}\left(1 - \frac{1}{\cos^2{x}}\right) =$

$\displaystyle \sin^2{x}\left(1 - \sec^2{x}\right) =$

$\displaystyle \sin^2{x}\left(-\tan^2{x}\right) = -\sin^2{x}\tan^2{x}$

$\displaystyle \sin^4{x} +2\sin^2{x}\cos^2{x} + \cos^4{x} =$

$\displaystyle \left(\sin^2{x} + \cos^2{x}\right)^2 = \, ?$
Woah, that is intense ... I some what understand these steps but it's a lot more advanced than the identities which we are currently being taught.

Is it possible if you can explain to me how you went from

$\sin^4{x} +2\sin^2{x}\cos^2{x} + \cos^4{x} =$

to

$\left(\sin^2{x} + \cos^2{x}\right)^2 = \, ?$

Thanks

5. ## re: Proving a Couple of Trig Identities

Originally Posted by Dante
Is it possible if you can explain to me how you went from

$\sin^4{x} +2\sin^2{x}\cos^2{x} + \cos^4{x} =$

to

$\left(\sin^2{x} + \cos^2{x}\right)^2 = \, ?$

Thanks
$\displaystyle a^2+b^2+2ab=(a+b)^2$