If sin θ + cosec θ = 2,

Given sinθ + cosecθ = 2

⇒ (sinθ + cosecθ)2 = 4

i.e sin2θ + cosec2θ + 2 sinθ  cosecθ = 4

i. e $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta+2 \sin \theta \frac{1}{\sin \theta}=4$

i. e $\sin ^{2} \theta+\operatorname{cosec}^{2} \theta=2$

Hence, the correct answer is option C.