Trig 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. $\csc x = \frac{1}{\sin x}$).

These help break down complicated $\tan$ and $\cot$ expressions to $\sin$ and $\cos$.

• $\tan x = \frac{\sin x}{\cos x}$
• $\cot x = \frac{\cos x}{\sin x}$

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. $\sin^2 x = 1 - \cos^2 x$).

Take a look at this if it doesn't make sense why this works¹⁾.

• $\sin(\frac{\pi}{2} - x) = \cos x$
• $\cos(\frac{\pi}{2} - x) = \sin x$
• $\tan(\frac{\pi}{2} - x) = \cot x$
• $\csc(\frac{\pi}{2} - x) = \sec x$
• $\sec(\frac{\pi}{2} - x) = \csc x$
• $\cot(\frac{\pi}{2} - x) = \tan x$

To remember this easier, think that $\cos$ and $\sec$ are the odd ones out²⁾. Otherwise, its just pulling the negative out.

• $\sin -x = -\sin x$
• $\tan -x = -\tan x$
• $\csc -x = -\csc x$
• $\cot -x = -\cot x$

• $\cos -x = \cos x$
• $\sec -x = \sec x$

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.

• $\sin(x \pm y) = \sin x\cos y \pm \cos x\sin y$
• $\cos(x \pm y) = \cos x\cos y \mp \sin x\sin y$
• $\tan(x \pm y) = \frac{\tan x \pm \tan y}{1 \mp \tan x\tan y}$

Cosine has alot of variations. Why? I have no fucking clue!

• $\sin 2x = 2\sin x\cos x$
• $\cos 2x = \cos^2 x - \sin^2 x = 2\cos^2 x - 1 = 1 - 2\sin^2 x$
• $\tan 2x = \frac{2\tan x}{1 - \tan^2 x}$

As the title says.

• $\sin^2 x = \frac{1 - \cos 2x}{2}$
• $\cos^2 x = \frac{1 + \cos 2x}{2}$
• $\tan^2 x = \frac{1 - \cos 2x}{1 + \cos 2x}$³⁾

The signs of $\sin \frac{x}{2}$ and $\cos \frac{x}{2}$ depends on the quadrant where $\frac{x}{2}$ lies.

• $\sin \frac{x}{2} = \pm \sqrt \frac{1 - \cos x}{2}$
• $\cos \frac{x}{2} = \pm \sqrt \frac{1 + \cos x}{2}$
• $\tan \frac{x}{2} = \frac{1 - \cos x}{\sin x} = \frac{\sin x}{1 + \cos x}$

No damn clue what these are used for, on the sheet so here it goes.

• $\sin x + \sin y = 2\sin(\frac{x + y}{2})\cos(\frac{x - y}{2})$
• $\sin x - \sin y = 2\cos(\frac{x + y}{2})\sin(\frac{x - y}{2})$
• $\cos x + \cos y = 2\cos(\frac{x + y}{2})\cos(\frac{x - y}{2})$
• $\cos x - \cos y = -2\sin(\frac{x + y}{2})\sin(\frac{x - y}{2})$

Again, i dunno just ctrl-c and ctrl-v.

• $\sin x\sin y = \frac{1}{2}(\cos(x - y) - \cos(x + y))$
• $\cos x\cos y = \frac{1}{2}(\cos(x - y) + \cos(x + y))$
• $\sin x\cos y = \frac{1}{2}(\sin(x + y) + \sin(x - y))$
• $\cos x\sin y = \frac{1}{2}(\sin(x + y) - \sin(x - y))$