Do you use logarithm and trigonometric rules by heart? That is technically no primary mathematical question, but I am really interested in that: I am able to prove that , e.g. $\cos(A + B) = \cos A\cos B - \sin A \sin B$ or that $\log(A)+ \log(B) = \log(AB)$. I also understand why this is.
But at the end of the day I simply have to memorise these rules. It's not like that I look at a calculation including trig-functions or logarithms and "have a natural feeling" for the calculation like dividing, multiplying, factorizing and so on, where I kinda "see" the result and the steps. 
So here's my question: I am always a little bit confused if I simply do not have enough routine in using these operations or if it is just the usual case, that one does not simply "see, or have a natural feeling" for these calculations ?  
Because whenever I see a professor or any tutorials dealing with them it seems like they get these results like they're "obviously to see" .
Both examples are randomly chosen, there are plenty of others related especially to these topics.
 A: To gain a "natural feeling" with some identities I've experienced that for me is not sufficient to know the proof of it. I've have to use it in solving excersises and problems to develop a fluently usage. Often It might not be sufficient. 
You can consider the definition in terms of exponential function of trigonometric functions from Euler's identity 
$$e^{ix}=\cos(x)+i\sin(x)$$
from that you can say that 
$$\cos(x)=\frac{e^{ix}+e^{-ix}}{2}, \qquad \sin(x)=\frac{e^{ix}-e^{-ix}}{2i}$$
from these identities you are able to derive known trig identities (e.g. sum and duplication).
$$\sin(2x)=\frac{e^{2ix}-e^{-2ix}}{2i}=\frac{(e^{ix}-e^{ix})(e^{ix}+e^{-ix})}{2i}=2\sin(x)\cos(x)$$
I think also that this is a more general learning problem: some mathematical stataments are unintuitive also after reading the proof of it. A necessary condition to know deeply a theorem is haved experienced with every element of its proof, they have to appears sufficiently clear to your mind.
Take a look to this question and respective answer!
A: There's probably a subtle difference between the recognition you acquire from the tactical use and reuse of a rule to solve problem and the ability to instantly find an explanation for it in terms of more basic rules. Then probably only the latter qualifies as intuition; and so practicing proofs, like you've been doing, is the best way of having explanations coming to mind quickly. 
But, just as when acquiring vocabulary in a foreign language, you probably have to forget and reconstruct proofs several times to acquire familiarity.
Some proofs are easier to recall than others, as I've found.
For example, you could prove the angle sum formulae through Euclidian geometry, through analytic geometry, or by considering that multiplying rotation matrices amounts to adding angles. It's this last approach I've found easiest to remember.
$\left( \begin{array}{c} c_1 & -s_1 \\ s_1 & c_1 \end{array} \right) \times \left( \begin{array}{c} c_2 & -s_2 \\ s_2 & c_2 \end{array} \right) =
\left( \begin{array}{c} c_1 c_2 - s_1 s_2 & \ldots \\ s_1 c_2 + c_1 s_2 & \ldots \end{array} \right)$
(Incidentally, the "row by column" approach to matrix multiplication is more tactical than intuitive. But you've got to pick when to function in recognition mode and in intuition mode depending on the needs of the moment.)
A: Once you've worked with these rules a lot, they will become "natural" - to me, the logarithm rules just "feel right" now, but when I first learned them they didn't. In the meantime, though, the thing to bear in mind is - in my opinion - that knowing the formula isn't actually important. You should know that the formula exists. Aim to start to recognize that, for example, $\log(AB)$ can be replaced with something to get rid of the product. Once you've spotted that that can be done, then you can search your memory or your notes for the exact formula. To take a personal example: I honestly don't know some of the trig identities involving sums of angles. Instead, I just remember that there is a formula that allows me to turn $\sin{A}\cos{B}$ into a sum of two trig functions - which is enormously helpful for integration.
Also, as general advice: Bear in mind that your instructor and any online tutoring videos you watch usually prepare the problems they do extensively ahead of time. That's why you never see them make arithmetic errors or do the problem wrong - and it's why they never seem to have to think about what they're doing. Don't expect yourself to be able to do the problem as easily.
A: Exponentiation.  This is probably the most important of the rules to know.
$(x^2)(x^3) = (x\cdot x\cdot x\cdot x\cdot x) = x^5\\
(a^x)(a^y) = a^{x+y}\\
f(x) = e^x \implies f(x+y) = e^{x+y} = (e^x)(e^y) = (f(x))(f(y))$
Logarithms are the inverse of exponentiation.  And so the product rule reverses.
$\log ab = \log a + \log b$
The rest of the log rules are just consequences of that one.  In fact, if you know nothing else about logarithms, that is what you should know.
As for the trig functions.  Know how these functions relate to the unit circle and most of the trig identities flow from that.  I memorized the angle addition rules.
$\cos (x+y) = \cos x\cos y - \sin x\sin y\\
\sin(x+y) = \sin x\cos y + \cos x\sin y$
And so, I had to think, which one alternates sin and cosine functions in each term and which doesn't, and which one has a minus sign in it.
Worth knowing that $\sin x$ is a odd function...i.e. $\sin -x = - \sin x$ and $\cos x$ is an even function $\cos -x = \cos x$
There is a connection between the trig-functions and complex numbers.  $(\cos x + i \sin x)(\cos y + i\sin y) = \cos (x+y) + i\sin (x+y)$  And, so if you know this, then the derivation of the angle addition rules is obvious.
There is also a connection between the trig functions and the exponential function, but that might be one step too far for right now.
The rest of the identities and rules flow from this small set.  I never tried to memorize them, but I can look at $\cos a \cos b$ and know that there must be an identity that relates it to $\cos (a+b)$ and after thinking for a few seconds can derive the of the rule will come to me.
