If $(1+\tan 1^{\circ})(1+\tan 2^{\circ}) \ldots (1+\tan 45^{\circ})=2^n$,then $n=$

If $\theta+\phi=\alpha$ and $\tan \theta=k \tan \phi$ (where $k>1$),then the value of $\sin (\theta-\phi)$ is

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:

$\operatorname{cosec} 48^{\circ}+\operatorname{cosec} 96^{\circ}+\operatorname{cosec} 192^{\circ}+\operatorname{cosec} 384^{\circ} = $

When $\frac{\sin 9 \theta}{\cos 27 \theta}+\frac{\sin 3 \theta}{\cos 9 \theta}+\frac{\sin \theta}{\cos 3 \theta}=k(\tan 27 \theta-\tan \theta)$ is defined,then $k=$

The value of $\cos \frac{2 \pi}{15} \cos \frac{4 \pi}{15} \cos \frac{8 \pi}{15} \cos \frac{14 \pi}{15}$ is :