State whether the following is true or false. Justify your answer.
$\sin (A+B) = \sin A + \sin B$

If $4 \tan \theta = 3,$ then $\left(\frac{4 \sin \theta - \cos \theta}{4 \sin \theta + \cos \theta}\right)$ is equal to

Write 'True' or 'False' and justify your answer.
$\tan \theta$ increases faster than $\sin \theta$ as $\theta$ increases for $0^\circ < \theta < 90^\circ$.

If $\angle B$ and $\angle Q$ are acute angles such that $\sin B = \sin Q$,then prove that $\angle B = \angle Q$.

Write 'True' or 'False' and justify your answer.
$\cos \theta = \frac{a^{2} + b^{2}}{2ab}$,where $a$ and $b$ are two distinct numbers such that $ab > 0$.

State whether the following are true or false. Justify your answer.
$(i)$ The value of $\tan A$ is always less than $1$.
$(ii)$ $\sec A = \frac{12}{5}$ for some value of angle $A$.