What are some must-know trig identities? I'm having a hard time with trigonometry. Not in the sense that I can't solve problems, but that I don't understand what I'm using. I see a trig identity and just sort of accept that it works without necessarily seeing how it was derived. But most of the time when I see the equivalence I have no idea how they got there.
Are there a few "core" identities from which the rest can be derived? Or am I barking up the wrong tree and just need to suck it up and memorize?
 A: Learn and remember the basic geometric definition of trigonometric functions, notably


*

*$\cos x$ and $\sin x$ are the coordinates of the point M on the
trigonometric circle

*$\tan x$ is the y coordinate of the intersection between the vertical
line from (1,0) and the line OM

*...


and so on.

From here the foundamental relationships


*

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

*$\tan \theta=\frac{\sin \theta}{\cos \theta}$

*...


and so on.
The basic values for the basic angles for all trigonometric functions. Note that it suffice to memorize the important values in the first quadrant/octant and then obtain the others by symmetry and basic trigonometric identities.

Refer to the wiki List of trigonometric identities as summary.
A: 1) Learn the definitions of $$sin(x), cos(x), tan(x), cot(x),...$$
2) Learn basic relation between these functions, such as $$sin^2(x)+cos^2(x)=1, sec^2(x) = 1+ tan^2(x),...$$ 
3) Learn addition formulas such as $$sin(a+b)=sin(a)cos(b)+cos(a)sin(b),...$$
4) Learn double angle formulas, such as $$sin(2a) = 2sin(a)cos(a),....$$
5) learn trig functions of $ \pi$ , $\pi /2,$, $ \pi /6,$.....    
