The value of $\sqrt{3} \cot 20^{\circ}-4 \cos 20^{\circ}$ is equal to

If $A+B=\frac{\pi}{2}$ then the maximum value of $\cos \mathrm{A} \cdot \cos \mathrm{B}$ is

If $\tan \mathrm{A}=\frac{1}{\sqrt{x\left(x^2+x+1\right)}}, \tan \mathrm{B}=\frac{\sqrt{x}}{\sqrt{x^2+x+1}}$ and $\tan \mathrm{C}=\sqrt{x^{-1}+x^{-2}+x^{-3}}$ then

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