What is the significance of the general solution of trigonometric equations?
$\sin 5x+\sin x= \sin 4x+\sin 2x$ on simplification becomes $\cos 2x=\cos x$ which can be transformed to $\sin 2x=\sin x$. The general solution of both forms are different.
for $\cos 2x=\cos x$ , it is $2n\pi /3$ and for $\sin 2x=\sin x$, it is $(2n+1)\pi /3$.
Please tell me the significance of general solution.
In general, by the definition on the unit circle and for symmetry we have that
$$\cos \theta = \cos \alpha \iff \theta=\alpha +2k\pi \quad \lor \quad \theta=-\alpha +2k\pi$$
therefore in this case
$$\cos 2x = \cos x \iff 2x=x +2k\pi \quad \lor \quad 2x=-x +2k\pi$$
which leads to the solutions
$$x= \frac23k\pi \quad \lor \quad x= 2k\pi$$
which corresponds to $x= \frac23k\pi$ with $k\in \mathbb Z$.
Refer to the related
As an alternative, in this case we can also use that
$$\cos 2x=\cos x \iff 2 \cos^2 x- \cos x -1=0$$
which leads to $\cos x=-\frac12$ and $\cos x=1$.
$$\sin 5x+\sin x=\sin 4x +\sin 2x$$
$$2\sin 3x\cos 2x=2\sin 3x \cos x$$
$$\cos 2x=\cos x$$
$$2x=2n\pi \pm x$$
$$x=\frac{2n\pi}{3},2n\pi$$
You would have also to consider the other case when
$$\sin 3x=0$$
$$3x=n\pi$$
$$x=\frac{n\pi}{3}$$
where $n \in Z$
$\sin \theta$ and $\cos \theta$ are periodic functions. They regularly repeat after an interval.
This arose the need of general solution.