Let $f_k(x) = \frac{1}{k}(\sin^k x + \cos^k x)$ for $k = 1, 2, 3, ...$. Then for all $x \in R$,the value of $f_4(x) - f_6(x)$ is equal to

If $\cos A = m \cos B,$ then

If $\frac{\tan(A-B)}{\tan A} + \frac{\sin^{2}C}{\sin^{2}A} = 1,$ where $A, B, C \in (0, \frac{\pi}{2})$, then:

Let $\theta, \phi \in [0, 2\pi]$ be such that $2 \cos \theta(1-\sin \phi) = \sin^2 \theta \left(\tan \frac{\theta}{2} + \cot \frac{\theta}{2}\right) \cos \phi - 1$,$\tan (2\pi - \theta) > 0$ and $-1 < \sin \theta < -\frac{\sqrt{3}}{2}$. Then $\phi$ cannot satisfy

If $y^2+z^2=a y z$,$z^2+x^2=b x z$,and $x^2+y^2=c x y$,then the value of $\frac{x z}{y^2}+\frac{y^2}{z x}$ is

Let the set of all $a \in \mathbb{R}$ such that the equation $\cos 2x + a \sin x = 2a - 7$ has a solution be $[p, q]$ and $r = \tan 9^{\circ} - \tan 27^{\circ} - \frac{1}{\cot 63^{\circ}} + \tan 81^{\circ}$,then $pqr$ is equal to ....................