Let's say that you were trying to prove the following equation: sin2x/(1+cos2x) = tanx

I know that the equation is an identity since when I simplify the left-hand side, it is equivalent to the right-hand side.

But let's say that I replaced x with pi/4, and still showed that both sides were equivalent - would using only one value be enough to prove that the equation is an identity? I would think that by proving the equation using only one value (e.g. pi/4), would not be enough to prove that the equation is an identity. I thought that x takes into account all values, so it is more accurate than merely subbing in pi/4.

The definition of a trigonometric identity is as follows:

I want to place emphasis on the fact that the definition says that the solution set for trigonometric identities is all real numbers (and not including the solutions that are undefined). So with that being said, if x was only equal to pi/4, then how could the equation possibly be an identity?