If the real part of the complex number \((1-\cos \theta +2i\sin \theta {)}^{-1}\) is \(\frac{1}{5}\) for \(\theta \in (0…

If the real part of the complex number \((1-\cos \theta +2i\sin \theta {)}^{-1}\) is \(\frac{1}{5}\) for \(\theta \in (0,\pi )\), then the value of the integral \({\int }_{0}^{\theta }\sin xdx\) is equal to:

[JEE Main 2021, 20 Jul (Shift 2)]