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

The value of $\frac{\sin \theta + \sin 3 \theta}{\cos \theta + \cos 3 \theta}$ is

If $\cos \alpha + \cos \beta = \frac{1}{3}$ and $\sin \alpha + \sin \beta = \frac{1}{4}$,then $\cos (\alpha + \beta) = $

Prove that $\frac{\cos 9x - \cos 5x}{\sin 17x - \sin 3x} = -\frac{\sin 2x}{\cos 10x}$

If $\cos (A + B) = \alpha \cos A \cos B + \beta \sin A \sin B,$ then $(\alpha, \beta) =$

Let $\cos(\alpha+\beta)=-\frac{1}{10}$ and $\sin(\alpha-\beta)=\frac{3}{8}$ where $0 < \alpha < \frac{\pi}{3}$ and $0 < \beta < \frac{\pi}{4}$. If $\tan 2\alpha=\frac{3(1-r\sqrt{5})}{\sqrt{11}(s+\sqrt{5})}$,where $r, s \in N$,then $r+s$ is equal to . . . . . .