How to deduce trigonometric formulae like $2 \cos(\theta)^{2}=\cos(2\theta) +1$? Very important in integrating things like $\int \cos^{2}(\theta) d\theta$ but it is hard for me to remember them. So how do you deduce this type of formulae? If I can remember right, there was some $e^{\theta i}=\cos(\theta)+i \sin(\theta)$ trick where you took $e^{2 i \theta}$ and $e^{-2 i \theta}$. While I am drafting, I want your ways to remember/deduce things (hopefully fast).
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*About TPV's suggestion, how do you attack it geometrically??


$\cos^{2}(x) - \sin^{2}(x)=\cos(2x)$
plus $2\sin^{2}(x)$, then
$\cos^{2}(x)+\sin^{2}(x)=\cos(2x)+2\sin^{2}(x)$
and now solve homogenous equations such that LHS=A and RHS=B, where $A\in\mathbb{R}$ and $B\in\mathbb{R}$. What can we deduce from their solutions?

It might be useful to remember that $\cos^2 x$ oscillates twice as fast as $\cos x$. This is something that people who work with alternating current know very well; the effect (which is proportional to the square of the current) has twice the frequency. For example, a light bulb flickers at 100 Hz if the AC frequency is 50 Hz. This means that $\cos^2 x$ should be "something with $\cos 2x$".
Next, since $\cos x$ oscillates between -1 and 1, $\cos^2 x$ will oscillate between 0 and 1. The average around which the curve oscillates will be 1/2, and the amplitude will also be 1/2 (so that you reach down to 0 and up to 1 from the central level 1/2). Thus $\cos^2 x$ should be "1/2 + (1/2)*oscillating term".
Combining these two facts, it's not too hard to remember that
$$ \cos^2 x = \frac12 + \frac12 \cos 2x.$$
One has to be a little bit careful with the sign before the second term, but it must be plus if the formula is to hold when $x=0$.
If you choose the minus sign, you get the related formula
$$ \sin^2 x = \frac12 - \frac12 \cos 2x.$$
(So $\sin^2 x$ also oscillates around 1/2 with amplitude 1/2 and twice the frequency. Note that when you add the two formulas up, the oscillations cancel, and you get $\cos^2 x + \sin^2 x = 1/2+1/2 = 1$.)
