Math, how do we know if a substitution is true? For instance, in calculus we often do u-substitutions.  Quite often, we do trignometric substitutions to solve integrals.  For instance, if we have the following relation
$y=\sqrt{1-x^2}$
And we substitute $x = \sin u$, for $x \in [-1, 1]$; how do we know that our substitution is correct? If I graph $\sin u$ and $x$, it is clear these functions are very different.
How is there any relation at all? Why are we allowed to do substitutions like this?

Jason, when we make a substitution such as $u = sin(x)$, we are not really comparing two things and stating their equality, we are declaring that $u$ is $\textbf{another name}$ for $sin(x)$. The symbol $u$ has no inherent meaning, we are defining it to be a representation of the function $sin(x)$. It is just a symbol.
