Question

What is $\sin \left(4x\right)$ in terms of $\sin \left(2x\right)$ and $\cos 2x$?

Answer 1

Trignometric identies:

$\mathbf{\because }\mathbf{sin}\mathbf{}\mathbf{2}\mathbf{A}\mathbf{=}\mathbf{2}\mathbf{sin}\mathbf{}\mathbf{A}\mathbf{·}\mathbf{cos}\mathbf{}\mathbf{A}$

Let $A=2x$

$$\begin{gathered} \therefore \sin 2\left(2x\right)=2\sin 2x·\cos 2x \\ \Rightarrow \sin 4x=2\sin 2x·\cos 2x \end{gathered}$$

Hence, the value of $\mathbf{sin}\mathbf{}\mathbf{\left(}\mathbf{4}\mathbf{x}\mathbf{\right)}$ in terms of $\mathbf{sin}\mathbf{}\mathbf{\left(}\mathbf{2}\mathbf{x}\mathbf{\right)}$ and $\mathbf{cos}\mathbf{}\mathbf{2}\mathbf{x}$ is expressed as $\mathbf{sin}\mathbf{}\mathbf{4}\mathit{x}\mathbf{=}\mathbf{2}\mathbf{}\mathbf{sin}\mathbf{}\mathbf{\left(}\mathbf{2}\mathit{x}\mathbf{\right)}\mathbf{·}\mathbf{cos}\mathbf{}\mathbf{\left(}\mathbf{2}\mathit{x}\mathbf{\right)}$