If $\cos \theta - \sin \theta = \sqrt{2} \sin \theta,$ then $\cos \theta + \sin \theta$ is equal to

The number of all possible values of $\theta$,where $0 < \theta < \pi$,for which the system of equations
$(y+z) \cos 3\theta = (xyz) \sin 3\theta$
$x \sin 3\theta = \frac{2 \cos 3\theta}{y} + \frac{2 \sin 3\theta}{z}$
$(xyz) \sin 3\theta = (y+2z) \cos 3\theta + y \sin 3\theta$
have a solution $(x_0, y_0, z_0)$ with $y_0 z_0 \neq 0$,is

The value of $\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$ is :

$\frac{\sin 3A - \cos \left( \frac{\pi}{2} - A \right)}{\cos A + \cos (\pi + 3A)} = $

If $\sin \alpha + \cos \alpha = m$,then $\sin^6 \alpha + \cos^6 \alpha = $

If $f_n(x) = \frac{1}{2n} [\sin^{2n} x + \cos^{2n} x]$,then $f_1(x) + f_2(x) - f_3(x) =$