Let $a$ and $b$ be non-negative real numbers. If $\sin x + a \cos x = b$,then $|a \sin x - \cos x| = $

Let $\alpha = \frac{1}{\sin 60^{\circ} \sin 61^{\circ}} + \frac{1}{\sin 62^{\circ} \sin 63^{\circ}} + \dots + \frac{1}{\sin 118^{\circ} \sin 119^{\circ}}$. Then the value of $\left(\frac{\operatorname{cosec} 1^{\circ}}{\alpha}\right)^2$ is $....$

If $A$ and $B$ are acute angles satisfying $3 \cos^2 A + 2 \cos^2 B = 4$ and $\frac{3 \sin A}{\sin B} = \frac{2 \cos B}{\cos A}$,then $A + 2B =$

If $\tan A - \tan B = x$ and $\cot A - \cot B = y$,then $\cot (A - B) =$

If $\tan \theta + \sin \theta = a$ and $\tan \theta - \sin \theta = b$,then the values of $\cot \theta$ and $\operatorname{cosec} \theta$ are respectively

If $\sin \theta + \sin \phi = a$ and $\cos \theta + \cos \phi = b,$ then $\tan \frac{\theta - \phi}{2}$ is equal to