$\sqrt {2 + \sqrt {2 + 2\cos 4\theta } } = $

If $\cos (\alpha + \beta ) = \frac{4}{5}, \sin (\alpha - \beta ) = \frac{5}{13}$ and $\alpha, \beta$ lie between $0$ and $\frac{\pi}{4}$,then $\tan 2\alpha = $

The value of the following is $\cos 24^{\circ} + \cos 55^{\circ} + \cos 125^{\circ} + \cos 204^{\circ} + \cos 300^{\circ}$

If $x \sin^{2} 60^{\circ} - \frac{3}{2} \sec 60^{\circ} \tan^{2} 30^{\circ} + \frac{4}{5} \sin^{2} 45^{\circ} \tan^{2} 60^{\circ} = 0$,then $x$ is

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

Let $\alpha$ and $\beta$ be two real roots of the equation $(k+1) \tan ^{2} x-\sqrt{2} \lambda \tan x=(1-k)$ where $k(\neq-1)$ and $\lambda$ are real numbers. If $\tan ^{2}(\alpha+\beta)=50,$ then a value of $\lambda$ is :