If $A, B, C$ are angles of a triangle,then $\sin 2A + \sin 2B - \sin 2C$ is equal to

If $5 \tan \theta = 4$,then $\frac{5 \sin \theta - 3 \cos \theta}{5 \sin \theta + 2 \cos \theta} = $

Let $A_0 A_1 A_2 A_3 A_4 A_5$ be a regular hexagon inscribed in a circle of unit radius. Then the product of the lengths of the line segments $A_0 A_1$,$A_0 A_2$,and $A_0 A_4$ is

The value of $\frac{(\tan 69^{\circ} + \tan 66^{\circ})}{(1 - \tan 69^{\circ} \tan 66^{\circ})}$ is

Given that $\cos \left( \frac{\alpha - \beta}{2} \right) = 2\cos \left( \frac{\alpha + \beta}{2} \right)$,then $\tan \frac{\alpha}{2} \tan \frac{\beta}{2}$ is equal to

Let $n$ be a positive integer such that $\sin \frac{\pi }{2^n} + \cos \frac{\pi }{2^n} = \frac{\sqrt{n}}{2}.$ Then