If $x = \log \left[ \cot \left( \frac{\pi}{4} + \theta \right) \right]$, then the value of $\sinh x$ is

If $a = \sin \frac{\pi}{18} \sin \frac{5\pi}{18} \sin \frac{7\pi}{18}$ and $x$ is the solution of the equations $y = 2[x] + 2$ and $y = 3[x - 2]$,where $[x]$ denotes the greatest integer function of $x$,then $a$ is equal to:

If $A$ is in the third quadrant and $\tan A = \frac{\sqrt{7}}{3}$,then $18 - 16 \sin^2 \frac{A}{2} - 32 \sin \frac{A}{2} \sin \frac{5A}{2} = $

Let the range of the function $f(x) = 6 + 16 \cos x \cdot \cos \left(\frac{\pi}{3} - x\right) \cdot \cos \left(\frac{\pi}{3} + x\right) \sin 3x \cdot \cos 6x$,where $x \in R$,be $[\alpha, \beta]$. Then the distance of the point $(\alpha, \beta)$ from the line $3x + 4y + 12 = 0$ is:

If $\tan A$ and $\tan B$ are the roots of the quadratic equation $x^2-px+q=0$,then $\sin^2(A+B)$ is equal to

Let $a, b, c$ be the sides of a triangle. If $t$ denotes the expression $\frac{a^2+b^2+c^2}{ab+bc+ca}$,the set of all possible values of $t$ is