If $\cos \frac{\pi}{15} \cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{5 \pi}{15} \cos \frac{7 \pi}{15} \cos \frac{30 \pi}{15} = x$,then $\frac{1}{8x} =$

Let $a, b, c$ be three non-zero real numbers such that the equation $\sqrt{3} a \cos x + 2 b \sin x = c$,$x \in [-\frac{\pi}{2}, \frac{\pi}{2}]$ has two distinct real roots $\alpha$ and $\beta$ with $\alpha + \beta = \frac{\pi}{3}$. Then the value of $\frac{b}{a}$ is

If $A+B+C+D=2 \pi$,then $\cos A-\cos B+\cos C-\cos D=$

If $\tan A - \tan B = x$ and $\cot A - \cot B = y$,then $\cot (A - B) =$

$\operatorname{sech}^{-1}(\sin \theta)$ is equal to

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: