The value of $\cos 1^{\circ} \cos 2^{\circ} \cos 3^{\circ} \ldots \cos 179^{\circ} = $

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

$\sin^{2} \theta = \frac{(x+y)^{2}}{4xy}$ is possible only when

$3\,\left[ {{{\sin }^4}\,\left( {\frac{{3\pi }}{2} - \alpha } \right) + {{\sin }^4}\,(3\pi + \alpha )} \right] - 2\,\left[ {{{\sin }^6}\,\left( {\frac{\pi }{2} + \alpha } \right) + {{\sin }^6}(5\pi - \alpha )} \right] = $

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

If $\tan A = 2\tan B + \cot B,$ then $2\tan (A - B) = $