If $A + B + C = {270^o},$ then $\cos \,2A + \cos 2B + \cos 2C + 4\sin A\,\sin B\,\sin C = $

$\sin 12^\circ \sin 48^\circ \sin 54^\circ = $

$\frac{{\tan A + \sec A - 1}}{{\tan A - \sec A + 1}} = $

If $\alpha ,\,\,\beta ,\gamma ,\,\,\delta $ are the smallest positive angles in ascending order of magnitude which have their sines equal to the positive quantity $k$, then the value of $4\,\sin \frac{\alpha }{2} + 3\,\sin \frac{\beta }{2} + 2\,\sin \frac{\gamma }{2} + \sin \frac{\delta }{2}$ is equal to

If $\tan \beta = \cos \theta \tan \alpha ,$ then ${\tan ^2}\frac{\theta }{2} = $

If $\cos x + \cos y + \cos \alpha = 0$ and $\sin x + \sin y + \sin \alpha = 0,$ then $\cot \,\left( {\frac{{x + y}}{2}} \right) = $