If $\sin 2\theta + \sin 2\phi = 1/2$ and $\cos 2\theta + \cos 2\phi = 3/2$,then $\cos^2(\theta - \phi) = $

If $\sin \alpha$ and $\cos \alpha$ are the roots of the equation $ax^2 + bx + c = 0$ $(c \neq 0)$,then:

If $\tan \theta = \frac{1}{\sqrt{11}}$ and $0 < \theta < \frac{\pi}{2},$ then the value of $\frac{\operatorname{cosec}^{2} \theta - \sec ^{2} \theta}{\operatorname{cosec}^{2} \theta + \sec ^{2} \theta}$ is

If $\sin \left( x + \frac{4\pi}{9} \right) = a$ and $\frac{\pi}{9} < x < \frac{\pi}{3}$,then $\cos \left( x + \frac{7\pi}{9} \right)$ equals:

The angle of elevation of the top of an unfinished tower at a point $120\, m$ from its base is $45^{\circ}$. How much higher must the tower be raised so that its angle of elevation at the same point be $60^{\circ}$? (in $m$)

The value of $x$ for the maximum value of $\sqrt{3} \cos x + \sin x$ is .....$^o$