Consider the following equality:
$$ \sin (t + \pi) = \sin (t)$$
$$ t + 0.5\pi = t + k2\pi \space \text{or} \space t + 0.5\pi = \pi - t + k2\pi $$
Let's say you want to find the solutions on $[0, 2 \pi]$, what does the left expression mean:
$$ k2\pi = 0.5 \pi$$
What does this mean mathematically? Does it contribute to the solution set at all?
I will ignore the specific equations given in the question, because it seems the point is to understand why the term $k2\pi$ (or $n2\pi$ or $2n\pi$ or other variations) sometimes appears in an equation in the solution of a math problem.
Here is a simple example: find all $t \in \mathbb{R}$ such that $$ \sin(t) = -\frac{1}{2}. $$
Now, we know $\sin\left( \frac{\pi}{6} \right) = \frac{1}{2}$ and $\sin(-t) = -sin(t)$, so $t = -\frac{\pi}{6}$ is one obvious solution to this equation. But that is not the only value of $t$ that solves the equation. You could also set $t = -\frac{\pi}{6} + 2\pi$, or $t = -\frac{\pi}{6} + 4\pi$, or $t = -\frac{\pi}{6} - 2\pi$, or $t = -\frac{\pi}{6} - 34\pi$.
In fact there are many possible solutions. What do they all have in common? Every one can be written in the form $$t = -\frac{\pi}{6} + k2\pi \ \mbox{where $k$ is an integer.}$$
This is especially good to know in case there are some other conditions that $t$ must satisfy. For example, if you want $t \in [0, 2\pi)$, not just any $t \in \mathbb{R}$, then clearly the "solution" $-\frac{\pi}{6}$ is not acceptable. But $-\frac{\pi}{6} + 2\pi \in [0, 2\pi)$, so that solution is acceptable. The term $k2\pi$ lets us add or subtract $2\pi$ as many times as it takes to reach a value that satisfies all our conditions, if there is any such value.