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The domain of a function $f:X\to Y$ is the specified set $X$ of allowed inputs. A function must assign a value to every member of that set. Its codomain $Y$ separately specifies where the outputs are allowed to lie.

The formula alone need not determine the domain. The rule $x\mapsto x^2$ on the real numbers and its restriction to the nonnegative real numbers have different allowed inputs. The first is not injective, while the second is. Similarly, the real-valued rule $x\mapsto 1/x$ excludes zero from its domain of a function.

Changing a formula by cancellation can alter where it is defined. The principle preserve the domain when simplifying a function records why an extension across a cancelled denominator is a separate operation.

Where the original expression is defined, multiplication by $\cos x$ givesThus the graph repeats with period $\pi$ and has vertical asymptotesBetween consecutive asymptotes the extended cotangent branch decreases from $+\infty$ to $-\infty$, since $y^{\prime }=-1/(\cos x+\sin x{)}^2<0$. It passes through $(k\pi ,1)$ and $(\pi /4+k\pi ,1/2)$.

It is essential to preserve the domain when simplifying a function. At $x=\pi /2+k\pi$, the original tangent is undefined. The finite limiting value is zero, so the graph has holes at $(\pi /2+k\pi ,0)$, rather than actual zeros. The simplified expression extends across those points, but the original one does not.