TITLE: How to remember trig identities?
QUESTION [14 upvotes]: Suppose I have a trig function $T: \Bbb{R} \rightarrow \Bbb{R}$. I want to be able to derive four basic properties: 
$$T(x) \cdot T(y)$$
$$T(x) + T(y)$$
$$T(x+y)$$
$$T(cx)$$
where $c$ is some scalar. 
I know there are a bunch of identities: reciprocal, quotient, Pythagorean,  co-function, even-odd. And then some formulas: product-to-sum, sum-to-product, sum-difference, double angle, half-angle/power-reducing. Here is a list. That's a lot to memorize and some of them seem to overlap.  
Why do the four basic properties I mentioned  require so many identities to learn? It is a bit cumbersome. Is there an easier way to learn how to do the basic arithmetic operations with trig functions?

REPLY [10 votes]: I find that four suffice.
$$\cos^2 (x) + \sin^2(x) = 1 \tag{1}$$
$$\cos(a+b)=\cos(a)\cos(b)-\sin(a)\sin(b) \tag{2}$$
$$\sin(a+b)=\sin(a)cos(b)+\sin(b)\cos(a) \tag{3}$$
$$\text{trig}(x) = \text{cotrig}(\frac{\pi}{2}-x) \tag{4}$$


The Pythagorean identity $(1)$ is easy to manipulate. Divide through by $cos^2(x)$ alternatively by $sin^2(x)$ to find the other forms
$$1 + \tan^2(x) = sec^2(x) \tag{5}$$
$$\cot^2(x) +1 = csc^2(x) \tag{6}$$
For the angle addition formulas $(2)$ and $(3)$, we can apply odd and even identities to quickly derive the angle subtraction identities:
$$\cos(a+\color{red}{(-b)})=\cos(a)\cos(\color{red}{(-b)})-\sin(a)\sin(\color{red}{(-b)})$$
$$\cos(a-b)=\cos(a)\cos(b)+\sin(a)\sin(b) \tag{7}$$
and
$$\sin(a+\color{red}{(-b)})=\sin(a)cos(\color{red}{(-b)})+\sin(\color{red}{(-b)})\cos(a)$$
$$\sin(a-b)=\sin(a)cos(b)-\sin(b)\cos(a) \tag{8}$$
We can also let $a=b$ and substitute in $(2)$ and $(3)$ to find double angle forms
$$\cos(a+a)=\cos(a)\cos(a)-\sin(a)\sin(a)$$
$$\cos(2a)=\cos^2(a)-\sin^2(a) \tag{9}$$
$$\sin(a+a)=\sin(a)cos(a)+\sin(a)\cos(a)$$
$$\sin(2a)=2\sin(a)cos(a) \tag{10}$$
Combining $(9)$ with the Pythagorean identity $(1)$ gives two more.
$$\cos(2a)=1-2\sin^2(a) \tag{11}$$
$$\cos(2a)=2\cos^2(a)-1 \tag{12}$$
If you want a half angle formula, you may as well let $u=2a$ in the previous four identities. Just mind your squares and roots. What happening to tangent? Divide any related $sin$ by $cos$ to get what you need.
Let's not forget product to sum identities! If we take $(2)$ and $(7)$ and add the equation, we find
$$\cos(a+b) + \cos(a-b)=2\cos(a)\cos(b)$$
$$\cos(a)\cos(b) = \frac12 (\cos(a+b) + \cos(a-b)) \tag{13}$$
Similar combinations will give the remaining product to sum identities.
As for $(4)$, $trig(x) = cotrig(\frac{\pi}{2}-x)$, I'm referring to cofunction identities, which all have the same form. For example, $\sin(x) = \cos(\frac{\pi}{2}-x).$ That's essentially six more identities.
We have over twenty identities at our disposal now, including the few that I've mentioned but neglected to outright derive.