The maximum value of $4\sin^2 x + 3\cos^2 x$ is

If $A+B+C=\frac{\pi}{4}$,then $4 \cos \frac{A}{2} \cos \frac{B}{2} \cos \frac{C}{2}-\cos \frac{\pi}{8}=$

$a, b, c$ are the sides of a scalene triangle $ABC$. If angles $\alpha, \beta, \gamma$ lie between $0$ and $\pi$ such that $\cos \alpha = \frac{a}{b+c}, \cos \beta = \frac{b}{c+a}$ and $\cos \gamma = \frac{c}{a+b}$,then $\tan^2 \frac{\alpha}{2} + \tan^2 \frac{\beta}{2} + \tan^2 \frac{\gamma}{2} =$

The minimum value of $27^{\cos 2x} 81^{\sin 2x}$ is.......

If $\alpha, \beta, \gamma$ are positive numbers such that $\alpha + \beta = \pi$ and $\beta + \gamma = \alpha$,then $\tan \alpha$ is equal to - (where $\gamma \neq n\pi, n \in I$)

In $\Delta ABC$,the minimum value of $\frac{\sum \cot^2 \frac{A}{2} \cot^2 \frac{B}{2}}{\prod \cot^2 \frac{A}{2}}$ is