### Trigonometry

goals
If tan $\beta \, = \, \dfrac{\tan \, \alpha \, + \, \tan \, \gamma}{1 \, + \, \tan \, \alpha \,\tan \, \gamma},$ Prove that:  $\sin \, 2\beta \, = \, \dfrac{\sin \, 2\alpha \, + \, \sin \, 2\gamma}{1 \, + \, \sin \, 2\alpha \, \sin \, 2\gamma}$
$\tan\theta +\tan 2\theta =\tan 3\theta$.
A helicopter is flying at an altitude of $250m$ between two banks of a river. From the helicopter it is observed that the angles of a river. From the helicopter its is observed  that the angle of depression of two boats on the opposite banks are $45^o$ and $60^o$ respectively. Find the width of the river. Given your answer correct to nearest metre.
If A, B, C are the angles of a triangle then show that
i) $\sin A + \sin B + \sin C \le \frac{{3\sqrt 3 }}{2}$
II) ${\tan ^2}\frac{A}{2} + {\tan ^2}\frac{B}{2} + {\tan ^2}\frac{C}{2} > 1.$
Solve
$\tan(A + B + C) = \dfrac{\sum \tan A \prod \tan A}{1 - \sum \tan A \tan B}$