Q.Find the value of $\sin\frac{31\pi}{3}$.

Some questions don't ask you to prove a general identity or solve an equation -- they give a specific angle or expression and ask for its exact numeric value, using one of a handful of known formulas as the tool. This is a distinct skill from proving an identity holds for every angle: here the destination is a single number.

Sum/difference formulas, used to reach a non-standard angle by combining two standard ones. For example, $\sin 75^\circ$ isn't a standard angle, but $75^\circ = 45^\circ + 30^\circ$ is a sum of two that are: $\sin 75^\circ = \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ = \dfrac{\sqrt6+\sqrt2}{4}$.

Half-angle formulas, used when the target angle is exactly half of a standard one. For example, $22\tfrac{1}{2}^\circ = 45^\circ/2$, so $\tan 22\tfrac12^\circ$ is found via the half-angle identity $\tan(\theta/2) = \dfrac{1-\cos\theta}{\sin\theta}$ with $\theta = 45^\circ$, giving $\sqrt2 - 1$. …