Using the general solution for solve sin⁡x=sin⁡yWe know that sin⁡x=sin⁡y implies that x=2kπ+y or x=2kπ+π−y. If we want to solve sin⁡x=sin⁡x using this method, it gives x=2kπ+x or x=2kπ+π−x but it's obvious that solution is x∈R. Why this happens? There is a similar problem for cosx=cosx.

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Using the general solution for solve sin⁡x=sin⁡yWe know that sin⁡x=sin⁡y implies that x=2kπ+y  or  x=2kπ+π−y. If we want to solve sin⁡x=sin⁡x using this method, it gives x=2kπ+x  or  x=2kπ+π−x but it&#039;s obvious that solution is x∈R. Why this happens? There is a similar problem for cosx=cosx.

Allison Compton · Accepted Answer

The point is in the difference between an identity and an equation.    For example   (x+1)2=x2+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.

Using the general solution for solve \sin x=\sin y We know that \sin x=\sin y implie

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