The value of $x$ for the maximum value of $(\sqrt{3} \sin x + \cos x)$ is .....$^o$.

If $A+B=225^{\circ}$,then the value of $\frac{\cot A}{1+\cot A} \cdot \frac{\cot B}{1+\cot B}$ is equal to

If the range of $f(\theta) = \frac{\sin^4 \theta + 3 \cos^2 \theta}{\sin^4 \theta + \cos^2 \theta}$,$\theta \in R$ is $[\alpha, \beta]$,then the sum of the infinite $G.P.$,whose first term is $64$ and the common ratio is $\frac{\alpha}{\beta}$,is equal to...........

The minimum value of $\frac{\tan(x + \frac{\pi}{6})}{\tan x}$ is

If $\cos A = \cos B \cos C$ and $A + B + C = \pi,$ then the value of $\cot B \cot C$ is

The extreme values of $1 + 4 \sin \theta + 3 \cos \theta$ are