If $A(n) = \sin^n \alpha + \cos^n \alpha$,then $A(1) A(4) + A(2) A(5) =$

$\left( {1 + \cos \frac{\pi }{8}} \right)\,\left( {1 + \cos \frac{{3\pi }}{8}} \right)\,\left( {1 + \cos \frac{{5\pi }}{8}} \right)\,\left( {1 + \cos \frac{{7\pi }}{8}} \right) = $

If $\cos \alpha + \cos \beta = \frac{3}{2}$ and $\sin \alpha + \sin \beta = \frac{1}{2}$ and $\theta$ is the arithmetic mean of $\alpha$ and $\beta$,then $\sin 2\theta + \cos 2\theta$ is equal to

If $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right)$,then:

Let $X, Y, Z$ be respectively the areas of a regular pentagon,regular hexagon,and regular heptagon which are inscribed in a circle of radius $1$. Then,

If $\sin x + \sin y = 3(\cos y - \cos x),$ then the value of $\frac{\sin 3x}{\sin 3y}$ is