The maximum value of $3\cos \theta - 4\sin \theta $ is

If $5 \sin \theta = 3$,find the numerical value of $\frac{\sec \theta - \tan \theta}{\sec \theta + \tan \theta}$.

If $\cos \theta = -\frac{\sqrt{3}}{2}$ and $\sin \alpha = -\frac{3}{5}$,where $\theta$ does not lie in the third quadrant and $\alpha$ lies in the third quadrant,find the value of $\frac{2 \tan \alpha + \sqrt{3} \tan \theta}{\cot^2 \theta + \cos \alpha}$.

If $\tan A + \sin A = m$ and $\tan A - \sin A = n,$ then $\frac{(m^2 - n^2)^2}{mn} = $

If ${\sin ^3}x \sin 3x = \sum\limits_{m = 0}^n {{c_m} \cos mx} $ where ${c_0}, {c_1}, {c_2}, \dots, {c_n}$ are constants and ${c_n} \ne 0$,then the value of $n$ is

If $\frac{\cos 3A + \sin 3A}{\cos A - \sin A} = 1 - K \sin 2A,$ the value of $K$ is