If $$A > 0,B > 0\,$$ and $$A + B = \pi /3,$$ then the maximum value of tan A tan B is _______.

Number of solutions of the equation $$\tan x + \sec x = 2\cos x\,$$ lying in the interval $$\left[ {0,2\pi } \right]$$ is:

$$ABCD$$ is a rhombus. Its diagonals $$AC$$ and $$BD$$ intersect at the point $$M$$ and satisfy $$BD$$ = 2$$AC$$. If the points $$D$$ and $$M$$ represent the complex numbers $$1 + i$$ and $$2 - i$$ respectively, then A represents the comp[lex number ..........or..........

Determine the smallest positive value of number $$x$$ (in degrees) for which $$$\tan \left( {x + {{100}^ \circ }} \right) = \tan \left( {x + {{50}^ \circ }} \right)\,\tan \left( x \right)\tan \left( {x - {{50}^ \circ }} \right).$$$