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

$\frac{\tan 52^{\circ} - \tan 38^{\circ}}{\tan 14^{\circ}} = $

Prove that: $\sin 3x + \sin 2x - \sin x = 4 \sin x \cos \frac{x}{2} \cos \frac{3x}{2}$

If $\alpha$ is in the $3^{\text{rd}}$ quadrant,$\beta$ is in the $2^{\text{nd}}$ quadrant such that $\tan \alpha = \frac{1}{7}$ and $\sin \beta = \frac{1}{\sqrt{10}}$,then find the value of $\sin(2\alpha + \beta)$.

If $(1 + \tan \theta )(1 + \tan \phi ) = 2$,then $\theta + \phi = \dots ^\circ$

$\cos^2 \left( \frac{\pi}{6} + \theta \right) - \sin^2 \left( \frac{\pi}{6} - \theta \right) = $