Maybe a little exaggerated?

This is a oral question if you apply CS inequality appropriately.. Let $\mathfrak S$ denote the given expression. Then by Cauchy-Schwarz-Inequality $\mathfrak S\geq \dfrac{(1+(\sqrt 2)^2+\cdots (\sqrt{100})^2)\cancel{\color{#D61F06}{[\sin^{2}(\theta_{1}) + \sin^{2}(\theta_{2}) + \cdots + \sin^{2}(\theta_{100})}}}{\cancel{\color{#D61F06}{\sin^{2}(\theta_{1}) + \sin^{2}(\theta_{2}) + \cdots + \sin^{2}(\theta_{100})}}}$ $\large =1+2+3+\cdots 100$ $\large =\dfrac{100(101)}{2}$ $\huge =\boxed{5050}$

Equality occurs when: $\dfrac{\sin \theta_1}{1}=\dfrac{\sin \theta_2}{\sqrt 2}=\cdots =\dfrac{\sin \theta _{100}}{\sqrt{100}}$

Just apply Cauchy inequality,for the numerator and the denominator will get cancelled,leaving $1+2+3+...100=5050$ .