If $\cos (A - B) = \frac{3}{5}$ and $\tan A \tan B = 2$,then

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

$\tan\, 20^{\circ} + \tan\, 40^{\circ} + \sqrt{3}\, \tan\, 20^{\circ} \tan\, 40^{\circ}$ is equal to

If $\cos (\theta+\phi)=\frac{3}{5}$ and $\sin (\theta-\phi)=\frac{5}{13}$, where $0 < \theta, \phi < \frac{\pi}{4}$, then $\cot (2 \theta)$ has the value:

The value of $(\cos \alpha+\cos \beta)^2+(\sin \alpha+\sin \beta)^2$ is

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 $\cos \left(\frac{\theta_1-\theta_2}{2}\right)=$