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Lets use an example:

$ \sin^2 \left(\dfrac{\pi}4x\right) = 1 $

I am at this point:

$ \frac{\pi}4 x=\frac{\pi}2 + k\cdot2\pi \quad\text{or}\quad \frac{\pi}4 x=-\frac{\pi}2 + k\cdot2\pi $

But then you have to merge the formulae into $ \frac{\pi}4 x = \frac{\pi}2 + k\cdot\pi $

This is not a hard example, but I have a LOT of trouble knowing when they are and when they aren't 'mergeable' , and how to easily figure out how to merge 2,3 or even 4 of these formulae into 1. How can I make this less troublesome? I am certain there is an easier way than just plain figuring it out in your head.. (like for example, I have no idea how to know quickly if for example $x =\frac{\pi}2 + k\cdot2\pi$ and $x=k\cdot2\pi$ are mergeable (my textbook indicates they aren't))

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    It is not as bad as π x 4 2012-09-15

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To avoid confusion, try introducing $t\equiv \frac{\pi x}{ 4}$.

You obtain $\sin^2t=1 \longrightarrow \sin t=\pm 1$, which are two easy to solve equations.

Once you have done this, substitute every solution in the equality that defines $t$ and solve for $x$.

About merging the solutions. First of all, you don't need to do that: saying that the full solution is the union of the partial solutions is perfectly fine. But I agree with you that sometimes it isn't so elegant.
Then you have to train your eye, so that you can notice patterns in the partial solutions, such that you can present them in a more compact way. Sometimes it's just unuseful to try this exercise (e.g. if you solve $\sin x = \frac12$, don't worry about merging $x=\frac{\pi}6+2k\pi$ and $x=\frac{5\pi}6+2k\pi$)!

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    $x=2+4k$ means that every element $x$ of the set $\{2+4k \;,\; k\in\mathbb{Z}\}$ is a solution of your equation. $x$ is the angle such that the sine (squared) of $\frac{\pi}4$ times $x$ is 1. So you can say that one of your solutions is (for $k=0$) $x=2\;\text{rad} \approx 104.6^{\circ}$. There's nothing wrong about not being a multiple of $\pi$, because $\pi$ isn't a unit of measurement of angles at all! The typical unit of measurement of angles is the radian, which is adimensional, while $\pi=\pi\; \text{rad} = 180^{\circ}$ is just a special angle. Hope it is clearer now! :)2012-09-15