For $\alpha, \beta \in \left(0, \frac{\pi}{2}\right)$,let $3 \sin (\alpha+\beta)=2 \sin (\alpha-\beta)$ and a real number $k$ be such that $\tan \alpha=k \tan \beta$. Then the value of $k$ is equal to :

$\frac{\sin^2 A - \sin^2 B}{\sin A \cos A - \sin B \cos B} = $

Suppose $\theta_1$ and $\theta_2$ are such that $(\theta_1-\theta_2)$ lies in the $3^{\text{rd}}$ or $4^{\text{th}}$ quadrant. If $\sin \theta_1+\sin \theta_2=-\frac{21}{65}$ and $\cos \theta_1+\cos \theta_2=-\frac{27}{65}$,then $\cos \left(\frac{\theta_1-\theta_2}{2}\right)=$

If $\sin \alpha = 1/\sqrt{5}$ and $\sin \beta = 3/5$,then $\beta - \alpha$ lies in the interval

$\tan 5x \tan 3x \tan 2x = $

$\tan 20^\circ + \tan 40^\circ + \sqrt{3} \tan 20^\circ \tan 40^\circ = $