Using the general solution for solve $\sin x = \sin y$

Question

We know that $\sin x = \sin y$ implies that $x = 2k\pi + y$ or $x = 2k\pi + \pi - y$. If we want to solve $\sin x = \sin x$ using this method, it gives $x = 2k\pi + x$ or $x = 2k\pi + \pi - x$ but it's obvious that solution is $x \in \mathbb{R}$. Why this happens? There is a similar problem for $\cos x = \cos x$.

Answer 1

The point is in the difference between an identity and an equation.

For example $$ (x+1)^2=x^2+2x+1$$ is an identity but $$(x+1)^2=16$$ is an equation.

The identity $$\sin x= \sin x $$ holds for every $x$ just like the identity $x=x$ and it does not require solving.

On the other hand the equation $$\sin x = \sin y$$ admits solutions because it is not true for all $x$ and $y$

We solve equations and find solutions because not every number satisfies an equation.

On the other hand we prove the identities and they are satisfied for all numbers.