Evaluate: $\sin 21^{\circ} \cos 9^{\circ}-\cos 84^{\circ} \cos 6^{\circ}$

If $\alpha_1, \alpha_2, \cdots, \alpha_n$ are in $A$.$P$. with common difference $\theta$, then the sum of the series $\sec \alpha_1 \sec \alpha_2 + \sec \alpha_2 \sec \alpha_3 + \cdots + \sec \alpha_{n-1} \sec \alpha_n = k(\tan \alpha_n - \tan \alpha_1)$, where $k=$

If $\sin A + \sin B = \frac{1}{2}$ and $\cos A + \cos B = 1$,then $\sin \left(\frac{A-B}{2}\right)$ equals

The equation $\frac{\sin^3 \theta - \cos^3 \theta}{\sin \theta - \cos \theta} - \frac{\cos \theta}{\sqrt{1 + \cot^2 \theta}} - 2 \tan \theta \cot \theta = -1$ holds true if:

If $\theta+\phi=\alpha$ and $\tan \theta=k \tan \phi$ (where $k>1$),then the value of $\sin (\theta-\phi)$ is

$\sin 5^{\circ} + \sin 10^{\circ} + \sin 15^{\circ} + \dots + \sin 350^{\circ} + \sin 355^{\circ}$ is equal to: