Let $\tan \alpha = \frac{a}{a+1}$ and $\tan \beta = \frac{1}{2a+1}$, then $\alpha + \beta$ is

If two angles $\alpha, \beta$ are such that $0 < \alpha, \beta < \frac{\pi}{4}$,$\sqrt{1+\cos 2 \alpha}=\frac{3}{\sqrt{5}}$ and $\frac{\sqrt{1-\cos 2 \beta}}{\sqrt{1+\cos 2 \beta}}=\frac{1}{7}$,then $(2 \alpha+\beta)=$

$\sin \left(\frac{\pi}{3}+x\right)-\cos \left(\frac{\pi}{6}+x\right) = $

If $\sin x + \sin y = \frac{1}{2}$ and $\cos x + \cos y = 1$,then $\tan(x + y) = $

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

Prove that $\frac{\sin x+\sin 3x}{\cos x+\cos 3x} = \tan 2x$.