If $\cos x - \sin x = \sqrt{a} \sin x$,then $a \sin x + \cos x - \sin x = $

If $\tan \theta - \cot \theta = a$ and $\sin \theta + \cos \theta = b,$ then ${({b^2} - 1)^2}({a^2} + 4)$ is equal to

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?

If $\sin \theta + \sin^2 \theta = 1$ and $\cos^{12} \theta + a \cos^{10} \theta + b \cos^8 \theta + c \cos^6 \theta + d = 0$,then:

If $\tan \alpha = \frac{x^2 - x}{x^2 - x + 1}$ and $\tan \beta = \frac{1}{2x^2 - 2x + 1}$ $(x \ne 0, 1)$,where $0 < \alpha, \beta < \frac{\pi}{2}$,then $\tan(\alpha + \beta)$ has the value equal to:

$\sin 12^\circ \sin 48^\circ \sin 54^\circ = $