If $x \sin \theta = y \sin \left( \theta + \frac{2\pi}{3} \right) = z \sin \left( \theta + \frac{4\pi}{3} \right)$,then:

If $x + y = 3 - \cos 4\theta$ and $x - y = 4 \sin 2\theta$,then:

$(1 - \tan 348^{\circ})(1 + \cot 417^{\circ})$ is equal to

If $\cos \alpha + \cos \beta = a$ and $\sin \alpha + \sin \beta = b$,then match the items given in List-$A$ with those of their values in List-$B$.

List-$A$	List-$B$
$(I)$ $\tan \left(\frac{\alpha + \beta}{2}\right) =$	$(a)$ $\frac{b}{a}$
$(II)$ $\cos (\alpha + \beta) =$	$(b)$ $\frac{2ab}{a^2 + b^2}$
$(III)$ $\sin (\alpha + \beta) =$	$(c)$ $\frac{2ab}{a^2 - b^2}$
$(IV)$ $\tan (\alpha + \beta) =$	$(d)$ $\frac{a^2 - b^2}{a^2 + b^2}$

Evaluate: $\sqrt{3} \operatorname{cosec} 20^{\circ} - \sec 20^{\circ}$

Let $A$ and $B$ denote the statements:
$A: \cos \alpha + \cos \beta + \cos \gamma = 0$
$B: \sin \alpha + \sin \beta + \sin \gamma = 0$
If $\cos (\alpha - \beta) + \cos (\beta - \gamma) + \cos (\gamma - \alpha) = -\frac{3}{2}$,then: