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

If the real part of the complex number \((1-\cos \theta+2 i \sin \theta)^{-1}\) is \(\frac{1}{5}\) for \(\theta \in(0, \pi)\), then the value of the integral \(\int_0^\theta \sin x d x\) is equal to :

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