If $\sin (A+B) \sin (A-B)+\cos (A+B) \cos (A-B)=\frac{1}{2}$ and $0 < B < \frac{\pi}{2}$,then $B=$

$\cos^2 \left( \frac{\pi}{6} + \theta \right) - \sin^2 \left( \frac{\pi}{6} - \theta \right) = $

If $\sin A = \frac{1}{\sqrt{10}}$ and $\sin B = \frac{1}{\sqrt{5}}$,where $A$ and $B$ are positive acute angles,then $A + B = $

The value of $\frac{\sin \theta + \sin 3 \theta}{\cos \theta + \cos 3 \theta}$ is

$\frac{1}{4} [\sqrt{3} \cos 23^\circ - \sin 23^\circ] = $

Let $\alpha, \beta$ be such that $\pi < (\alpha - \beta) < 3\pi$. If $\sin \alpha + \sin \beta = -\frac{21}{65}$ and $\cos \alpha + \cos \beta = -\frac{27}{65}$,then the value of $\cos \frac{\alpha - \beta}{2}$ is