Question : If $\alpha$ and $\beta$ are positive acute angles, $\sin (4\alpha -\beta )=1$ and $\cos (2\alpha +\beta)=\frac{1}{2}$, then the value of $\sin (\alpha +2\beta)$ is:

Question : If $(r\cos \theta -\sqrt{3})^{2}+(r\sin \theta -1)^{2}=0$, then the value of $\frac{r\tan \theta +\sec \theta}{r\sec \theta +\tan\theta}$ is equal to:

Question : If $\sin A-\cos A=\frac{\sqrt{3}-1}{2}$, then the value of $\sin A\cdot \cos A$ is:

Question : If $\frac{\cos\alpha}{\cos\beta}=a$, $\frac{\sin\alpha}{\sin\beta}=b$, then $\sin^{2}\beta$ is equal to:

Question : If $\theta$ is an acute angle and $\sin \theta \cos \theta=2 \cos ^3 \theta-\frac{1}{4} \cos \theta$, then the value of $\sin \theta$ is: