If $\frac{\sin(x + y)}{\sin(x - y)} = \frac{a + b}{a - b},$ then $\frac{\tan x}{\tan y}$ is equal to

If $\theta+\phi=\frac{\pi}{4}$, then $(1+\tan \theta)(1+\tan \phi)$ is equal to

Two ships leave a port from a point at the same time. One goes with a velocity of $3 \text{ km/h}$ along North-East making an angle of $45^{\circ}$ with the East direction and the other travels with a velocity of $4 \text{ km/h}$ along South-East making an angle of $15^{\circ}$ with the East direction. Then, the distance between the ships at the end of two hours is

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

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

Prove that $\sin (n+1) x \sin (n+2) x + \cos (n+1) x \cos (n+2) x = \cos x$.