If sin θ + cos θ = 1,

Given, $\sin \theta+\cos \theta=1$

By squaring both sides,

we get, $(\sin \theta+\cos \theta)^{2}=(1)^{2}=1$

i.e. $\sin ^{2} \theta+\cos ^{2} \theta+2 \sin \theta \cos \theta=1$

i. e. $1+2 \sin \theta \cos \theta=1$

Since $\sin ^{2} \theta+\cos ^{2} \theta=1$      (using identity)

1.e. $2 \sin \theta \cos \theta=0$

$\Rightarrow \sin 2 \theta=0$

Hence, the correct answer is option C.