If $y = 3\,sin\,x + 4\,cos\,x$,then find the maximum value of $y$.

If $\sin A = \frac{4}{5}$ and $\cos B = - \frac{12}{13},$ where $A$ and $B$ lie in the first and third quadrants respectively,then $\cos (A + B) = $

If $x \cos \theta = y \cos \left( \theta + \frac{2\pi}{3} \right) = z \cos \left( \theta + \frac{4\pi}{3} \right)$,then the value of $\frac{1}{x} + \frac{1}{y} + \frac{1}{z}$ is equal to

If $\sin x + \sin y = 3(\cos y - \cos x),$ then the value of $\frac{\sin 3x}{\sin 3y}$ is

If $12a + 5b = 9$,where $a, b \in R$,then the minimum value of $a^2 + b^2$ is -

$\frac{\cot \theta - \operatorname{cosec} \theta + 1}{\cot \theta + \operatorname{cosec} \theta - 1}$ is equal to