Question

Prove that $\sin 4A=4\sin A{\cos }^{3}A-4\cos A{\sin }^{3}A$.

Answer 1

Let us consider the problem:

$LHS=\sin 4A$

$RHS=4\sin A{\cos }^{3}A-4{\sin }^{3}A\cos A$

Let the consider $L.H.S$ $=\sin 4A$

$\Rightarrow$ $2\sin 2Acos2A$

$\Rightarrow$ $2\left[2\mathrm{sinA}\cos 2A\right]\left[{\cos }^{2}A-\sin 2A\right]$

$\Rightarrow$ $4\left[\sin {\cos }^{3}A-{\sin }^{3}A\cos A\right]$

$=RHS$