$\sin 15^\circ + \cos 105^\circ = $

If $A + B + C = 180^o,$ then the value of $(\cot B + \cot C)(\cot C + \cot A)(\cot A + \cot B)$ is

If the sides of a right-angled triangle are $\{\cos2\alpha + \cos2\beta + 2\cos(\alpha + \beta )\}$ and $\{\sin2\alpha + \sin2\beta + 2\sin(\alpha + \beta )\}$,then the length of the hypotenuse is:

If $90^{\circ} < \alpha < 180^{\circ}$,$\sin \alpha = \frac{\sqrt{3}}{2}$ and $180^{\circ} < \beta < 270^{\circ}$,$\sin \beta = -\frac{\sqrt{3}}{2}$,then $\frac{4 \sin \alpha - 3 \tan \beta}{\tan \alpha + \sin \beta} = $

If $A, B, C$ are the angles of a triangle,then $\sum \frac{\cot A + \cot B}{\tan A + \tan B} = $

The value of $\frac{\tan x}{\tan 3x}$,whenever defined,never lies between: