If triangle ABC is a right angled at A and $\tan \frac{\mathrm{B}}{2}$, $\tan \frac{\mathrm{C}}{2}$ are roots of the equation $a x^2+b x+c=0$, $\mathrm{a} \neq 0$, then

The value of $\cos 20^{\circ}+2 \sin ^2 55^{\circ}-\sqrt{2} \sin 65^{\circ}$ is

The maximum value of $\left(\cos \alpha_1\right) \cdot\left(\cos \alpha_2\right) \ldots .\left(\cos \alpha_n\right)$ under the constraints $0 \leq \alpha_1, \alpha_2, \ldots ., \alpha_n \leq \frac{\pi}{2}$ and $\left(\cot \alpha_1\right) \cdot\left(\cot \alpha_2\right) \ldots\left(\cot \alpha_n\right)=1$ is