Preserve the domain when simplifying a function

An algebraic simplification proves equality only where both original and transformed expressions are defined. If a cancelled factor vanishes at a point, the simplified expression may instead describe an extension to that point. For example,holds only when $\cos x\ne 0$ and $\cos x+\sin x\ne 0$. The right side is zero at $x=\pi /2+k\pi$, but the left side is undefined there. A graph of the original function must retain those holes. The extension may be useful, but it is a different function with a larger domain of a function.