# Thread: Verifying that this is an identity?

1. ## Verifying that this is an identity?

Trig and I are not very good friends...

$\displaystyle \frac{1 + tan^2x}{csc^2x} = tan^2x$

This is what I tried:

$\displaystyle \frac{1 + (sin^2x/cos^2x)}{(1/sin^2x)} = tan^2x$

$\displaystyle \frac{cos^2x + sin^2x}{cos^2x} * \frac{sin^2x}{1} = tan^2x$

$(sin^2x)(sin^2x) = tan^2x$

I don't know why it's not equaling up. D:

2. ## Re: Verifying that this is an identity?

Originally Posted by explodingtoenails
Trig and I are not very good friends...

$\displaystyle \frac{1 + tan^2x}{csc^2x} = tan^2x$

This is what I tried:

$\displaystyle \frac{1 + (sin^2x/cos^2x)}{(1/sin^2x)} = tan^2x$

$\displaystyle \frac{cos^2x + sin^2x}{cos^2x} * \frac{sin^2x}{1} = tan^2x$

...

I don't know why it's not equaling up. D:
From here, the top left is ONE.
Then you have (sin/cos)^2

3. ## Re: Verifying that this is an identity?

Thank you so much dude. I didn't know that cos^2x + sin^2x = 1.

4. ## Re: Verifying that this is an identity?

Originally Posted by explodingtoenails
Thank you so much dude. I didn't know that cos^2(x) + sin^2(x) = 1.
That's kind of important! From it, many (most?) identities are derived.

5. ## Re: Verifying that this is an identity?

Originally Posted by TheChaz
That's kind of important! From it, many (most?) identities are derived.
The most important identity really. By this I mean that as rings $\mathbb{R}[x,y]/(x^2-y^2-1)\cong \mathbb{R}[\cos(\theta),\sin(\theta)]$. In other words, if you take the ring $\mathbb{R}[x,y]$ and impose the conditions that $x^2+y^2=1$ then you really just have the ring of trigonometric polynomials $\mathbb{R}[\cos(\theta),\sin(\theta)]$.