Question

$2\sin \left(A\right){\cos }^3\left(A\right)-2{\sin }^3\left(A\right)\cos \left(A\right)=$

• A. $\sin \left(4A\right)$
• B. $\frac{1}{2}\sin \left(4A\right)$
• C. $\frac{1}{4}\sin \left(4A\right)$
• D. none of these

Answer 1

The correct option is B

$\frac{1}{2}\sin \left(4A\right)$

Explanation for correct option:

Given, $2\sin \left(A\right){\cos }^3\left(A\right)-2{\sin }^3\left(A\right)\cos \left(A\right)$

$=2\cos \left(A\right)\sin \left(A\right)\left({\cos }^2\left(A\right)-{\sin }^2\left(A\right)\right)$

Apply formula, $\mathit{s}\mathit{i}\mathit{n}\left(2x\right)\mathbf{=}\mathbf{2}\mathit{s}\mathit{i}\mathit{n}\left(x\right)\mathit{c}\mathit{o}\mathit{s}\left(x\right)\mathbf{}\mathbf{\&}\mathbf{}\mathit{c}\mathit{o}\mathit{s}\left(2x\right)\mathbf{=}\mathit{c}\mathit{o}{\mathit{s}}^{\mathbf{2}}\left(x\right)\mathbf{-}\mathit{s}\mathit{i}{\mathit{n}}^{\mathbf{2}}\left(x\right)$

$\begin{array}{rcl}& \Rightarrow & \left[2\cos \left(A\right)\sin \left(A\right)\right]\left[{\cos }^2\left(A\right)-{\sin }^2\left(A\right)\right]=\sin \left(2A\right)\cos \left(2A\right)\\ & =& \frac{1}{2}\left(2\sin \left(2A\right)\cos \left(2A\right)\right)\\ & =& \frac{1}{2}\sin \left(4A\right)\end{array}$

Therefore, correct answer is option $\left(B\right)$