Under complete dominance, an individual showing the dominant phenotype can have genotype $AA$ or $Aa$. A test cross distinguishes these possibilities by crossing the unknown individual with a homozygous recessive partner, $aa$.

Why use $aa$? Every gamete from the tester carries $a$, so each offspring directly reveals which allele came from the unknown parent.

Case 1: the unknown parent is $AA$

The cross is

$$AA\times aa.$$

The first parent produces only $A$ gametes and the tester produces only $a$ gametes, so every offspring is

$$Aa.$$

All offspring therefore show the dominant phenotype.

Case 2: the unknown parent is $Aa$

The cross is

$$Aa\times aa.$$

The heterozygous parent produces $A$ and $a$ gametes in equal proportions, while the tester still produces only $a$.

Therefore

$$P(Aa)=\frac12,\qquad P(aa)=\frac12.$$

The expected phenotype ratio is

$$1\text{ dominant}:1\text{ recessive}.$$

Observing any recessive offspring proves that the dominant-phenotype parent supplied an $a$ allele and therefore cannot have been $AA$.

Finite samples create uncertainty

If a heterozygous parent happens by chance to produce several dominant-phenotype offspring in a row, the cross can temporarily look like the $AA$ case.

For example, if the unknown parent is actually $Aa$, the probability that four offspring all show the dominant phenotype is

$$\left(\frac12\right)^4=\frac1{16}.$$

So "all observed offspring are dominant" is evidence for $AA$, but it is not absolute proof unless the underlying biological system or sample size makes alternative explanations negligible.

A test cross is therefore an inference procedure: use a genetically informative tester so that offspring phenotypes reveal otherwise hidden allele transmission from the unknown parent.