$2 \sin^2 \beta + 4 \cos(\alpha + \beta) \sin \alpha \sin \beta + \cos 2(\alpha + \beta) = $

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

Consider the following parametric equation of a curve: $x(\theta) = |\cos 4\theta| \cos \theta$ and $y(\theta) = |\cos 4\theta| \sin \theta$,where $0 \leq \theta \leq 2\pi$. Which one of the following graphs represents the curve?

$\tan 81^{\circ}-\tan 63^{\circ}-\tan 27^{\circ}+\tan 9^{\circ}$ equals to

The extreme values of $4 \cos \left(x^2\right) \cos \left(\frac{\pi}{3}+x^2\right) \cos \left(\frac{\pi}{3}-x^2\right)$ over $\mathbb{R}$ are

The value of $2 \sin(\frac{\pi}{8}) \sin(\frac{2\pi}{8}) \sin(\frac{3\pi}{8}) \sin(\frac{5\pi}{8}) \sin(\frac{6\pi}{8}) \sin(\frac{7\pi}{8})$ is: