====== Trig Identities ======
===== Reciprocal Identities =====
These are the ones you use to convert $\sin$, $\cos$, and $\tan$ to $\csc$, $\sec$, and $\cot$ respectively.
  * $\sin x = \frac{1}{\csc x}$
  * $\cos x = \frac{1}{\sec x}$
  * $\tan x = \frac{1}{\cot x}$

The same works vice-versa (e.g. $\csc x = \frac{1}{\sin x}$).

===== Quotient Identities =====
These help break down complicated $\tan$ and $\cot$ expressions to $\sin$ and $\cos$.
  * $\tan x = \frac{\sin x}{\cos x}$
  * $\cot x = \frac{\cos x}{\sin x}$

===== Pythagorean Identities =====
There are three essential ones to remember.
  * $\sin^2 x + \cos^2 x = 1$
  * $1 + \tan^2 x = \sec^2 x$
  * $1 + \cot^2 x = \csc^2 x$

You can rearrange these three as see fit to change practically any squared trig function to another (e.g. $\sin^2 x = 1 - \cos^2 x$).

===== Cofunction Identities =====
Take a look at [[https://www.desmos.com/calculator/vqoxhdcrv5|this]] if it doesn't make sense why this works((The period shift by $\frac{\pi}{2}$ lines up $\sin$ with $\cos$, or whatever other pair of trig functions)).
  * $\sin(\frac{\pi}{2} - x) = \cos x$
  * $\cos(\frac{\pi}{2} - x) = \sin x$
  * $\tan(\frac{\pi}{2} - x) = \cot x$
  * $\csc(\frac{\pi}{2} - x) = \sec x$
  * $\sec(\frac{\pi}{2} - x) = \csc x$
  * $\cot(\frac{\pi}{2} - x) = \tan x$

===== Even/Odd (Parity) Identities =====
To remember this easier, think that $\cos$ and $\sec$ are the odd ones out((See the [[math:unit_circle|unit circle]] for why)). Otherwise, its just pulling the negative out.
  * $\sin -x = -\sin x$
  * $\tan -x = -\tan x$
  * $\csc -x = -\csc x$
  * $\cot -x = -\cot x$

==== Cos and Sec ====
  * $\cos -x = \cos x$
  * $\sec -x = \sec x$

===== Sum and Difference Formulas =====
Take a note of the pos/neg sign, the upside down one indicates that it needs to be switched based off  whatever the pos/neg sign in the //original function// is. They keep the same sign if both signs are the same type.
  * $\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y$
  * $\cos(x \pm y) = \cos x\cos y \mp \sin x\sin y$
  * $\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x\tan y}$

===== Double Angle Formulas =====
Cosine has alot of variations. Why? I have no fucking clue!
  * $\sin 2x = 2\sin x\cos x$
  * $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$
  * $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$

===== Power-Reducing Formulas =====
As the title says.
  * $\sin^2 x = \frac{1 - \cos 2x}{2}$
  * $\cos^2 x = \frac{1 + \cos 2x}{2}$
  * $\tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x}$((Think back to the quotient identities for this one))

===== Half-Angle Formulas =====
The signs of $\sin \frac{x}{2}$ and $\cos \frac{x}{2}$ depends on the quadrant where $\frac{x}{2}$ lies.
  * $\sin \frac{x}{2} = \pm \sqrt \frac{1 - \cos x}{2}$
  * $\cos \frac{x}{2} = \pm \sqrt \frac{1 + \cos x}{2}$
  * $\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}$

===== Sum-To-Product Fomulas =====
No damn clue what these are used for, on the sheet so here it goes.
  * $\sin x + \sin y = 2\sin(\frac{x + y}{2})\cos(\frac{x - y}{2})$
  * $\sin x - \sin y = 2\cos(\frac{x + y}{2})\sin(\frac{x - y}{2})$
  * $\cos x + \cos y = 2\cos(\frac{x + y}{2})\cos(\frac{x - y}{2})$
  * $\cos x - \cos y = -2\sin(\frac{x + y}{2})\sin(\frac{x - y}{2})$

==== Product-To-Sum Formulas ====
Again, i dunno just ctrl-c and ctrl-v.
  * $\sin x\sin y = \frac{1}{2}(\cos(x - y) - \cos(x + y))$
  * $\cos x\cos y = \frac{1}{2}(\cos(x - y) + \cos(x + y))$
  * $\sin x\cos y = \frac{1}{2}(\sin(x + y) + \sin(x - y))$
  * $\cos x\sin y = \frac{1}{2}(\sin(x + y) - \sin(x - y))$