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:
Of course, on a unit circle, this works out to
We also know that on a unit circle:
Thus on a unit circle:
Note can also square it here:
Can also write it as follows:
From this we can easily derive:
Again:
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 \(cos^2\theta\)
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:
The rest are corrolaries you can do as needed