What is the maximum value of the functions $f(x) = \frac{1}{\tan x+\cot x},$ where $0 < x < \frac{\pi}{2}?$

If sinθ and cosθ are the roots of the equation ax² + bx + c = 0, then which one of the following is correct?

If sin(A + B) = 1 and 2 sin(A - B) = 1, where 0 < A, B < $\frac{\pi}{2}$, then what is tan A ∶ tan B equal to?

Let sin x + sin y = cos x + cos y for all x, y ∈ ℝ. What is $\rm tan \left(\frac{x}{2}+\frac{y}{2}\right)$ equal to?