If $0 < A < B < \frac{\pi}{4}$,$\cos (A+B) = \frac{11}{61}$ and $\sin (A-B) = \frac{24}{25}$,then $\sin 2A + \sin 2B = $

If $\tan \left( \frac{\pi }{4} + \theta \right) + \tan \left( \frac{\pi }{4} - \theta \right) = \lambda \sec 2\theta$,then $\lambda$ =

If $\frac{1-\tan \theta}{1+\tan \theta}=\frac{1}{\sqrt{3}}$,where $\theta \in \left(0, \frac{\pi}{2}\right)$,then $\theta=$

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 $m \tan (\theta - 30^\circ) = n \tan (\theta + 120^\circ)$,then $\frac{m + n}{m - n} = $

If $\cos (\alpha + \beta) = \frac{3}{5}$,$\sin (\alpha - \beta) = \frac{5}{13}$ and $0 < \alpha, \beta < \frac{\pi}{4}$,then $\tan (2\alpha)$ is equal to