# Even and Odd and Pythagorean Identities
(Trig 10 Professor Leonard - Precalc 83 )

## Even and Odd Identities

What even means -- symetric about the Y access (as output). Therefore opposite inputs (on the x axis, therefore, positive and negative) give you equal outputs.  i.e.: 
$f(-\theta) = f(\theta)$.  Odd functions have symettry around the origin, so can rotate graph 180 degrees, so opposite inputs give you opposite outputs. I.e. $f(-\theta) = - f(\theta)$.  Remember x is $cos(\theta),$ and y maps to $sin(\theta)$  Look at positive and negative $ \frac{pi}{6}:  \frac{pi}{6} = (\frac{\sqrt(3)}{2}, \frac{1}{2}), -pi/6 = (\frac{\sqrt(3)}{2}, \frac{1}{2})$

**Summary:**

Even:  cos, sec  
Odd: sin, csc, tan, cot  

Stands to reason because opposite angles are flipped accross x angles, and cos relates to y so it doesn't change.

|Even functions|Odd functions|
|----|---|
|$cos(- \theta) = cos(\theta)$|$sin(- \theta) = - sin(\theta)$|
|$sec(-\theta) = sec(\theta)$|$csc( -\theta) = -csc(\theta)$|
||$tan(-\theta) = -tan(\theta)$|
||$cot(-\theta) = -cot(\theta)$|

## Pythagorean Identities

Formula of circle:

$$ x^2 + y^2 = r^2 $$

Of course, on a unit circle, this works out to

$$ x^2 + y^2 = 1 $$

We also know that on a unit circle:

$$ cos(\theta) = x\space;\space sin(\theta) = y $$ 

Thus on a unit circle:

$$ (cos\space\theta)^2 + (sin\space\theta)^2 = 1 $$

Note can also square it here:

$$ cos^2\space\theta + sin^2\space\theta = 1 $$

Can also write it as follows:

$$ sin^2\space\theta +cos^2\space\theta = 1 $$

From this we can easily derive:

$$ sin^2\space\theta = 1 - cos^2\space\theta \\
	cos^2\space\theta = 1 - sin^2\space\theta
$$
Again:

$$ sin^2\theta +cos^2\theta = 1 $$

Can divide by $sin^2\theta$ or $cos^2\theta$:

### First divide by $sin^2\theta$

$ \frac{sin^2\theta}{sin^2\theta} + 
\frac{cos^2\theta}{sin^2\theta} = \frac{1}{sin^2\theta}  \\
So... \\
1 + \frac{cos^2\theta}{sin^2\theta} = \frac{1}{sin^2\theta} \\
so \\
1 + cot^2\theta  = csc^2\theta \\
\\ or \\
cot^2\theta = csc^2\theta -1 \\
or \\
csc^2\theta - cot^2\theta = 1 \\
$

### Next divide by $cos^2\theta$


$$ \frac{sin^2\theta}{cos^2\theta} + \frac{cos^2\theta}{cos^2\theta} = \frac{1}{cos^2\theta} $$

So...

$ 
tan^2\theta + 1 = sec^2\theta \\
or \\
tan^2\theta +  = sec^2\theta -1 \\
or \\
sec^2\theta - tan^2\theta = 1
$

## Three main ones to remember are below:

$$
sin^2\space\theta +cos^2\space\theta = 1 \\
\space\\ 
cot^2\theta + 1 = csc^2\theta \\
\space\\ 
tan^2\theta + 1 = sec^2\theta
$$

The rest are corrolaries you can do as needed