Question

$2\cos \left(x\right)-\cos \left(3x\right)-\cos \left(5x\right)=$

• A. $16{\cos }^3\left(x\right){\sin }^2\left(x\right)$
• B. $16{\sin }^3\left(x\right){\cos }^2\left(x\right)$
• C. $4{\cos }^3\left(x\right){\sin }^2\left(x\right)$
• D. $4{\sin }^3\left(x\right){\cos }^2\left(x\right)$

Answer 1

The correct option is A

$16{\cos }^3\left(x\right){\sin }^2\left(x\right)$

Explanation for correct option :

Simplify using standard trigonometric identities

Given, $2\cos \left(x\right)-\cos \left(3x\right)-\cos \left(5x\right)$

. $$\begin{gathered} =2\cos \left(x\right)-\left(\cos \left(3x\right)+\cos \left(5x\right)\right) \\ =2\cos \left(x\right)-\left(2\cos \left(\frac{3x+5x}{2}\right)\cos \left(\frac{3x-5x}{2}\right)\right)\mathbf{[}\mathbf{\because }\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{A}\mathbf{\right)}\mathbf{+}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\mathbf{B}\mathbf{\right)}\mathbf{=}\mathbf{2}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\frac{\mathbf{A}\mathbf{+}\mathbf{B}}{\mathbf{2}}\mathbf{\right)}\mathbf{c}\mathbf{o}\mathbf{s}\mathbf{\left(}\frac{\mathbf{A}\mathbf{-}\mathbf{B}}{\mathbf{2}}\mathbf{\right)}\mathbf{]} \\ =2\cos \left(x\right)-\left(2\cos \left(4x\right)\cos \left(x\right)\right) \\ =2\cos \left(x\right)\left[1-\cos \left(4x\right)\right] \\ =2\cos \left(x\right)\left(2{\sin }^2\left(2x\right)\right)\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{1}\mathbf{-}\mathbf{c}\mathbf{o}\mathbf{s}\left(2x\right)\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}{\mathbf{n}}^{\mathbf{2}}\left(x\right)\mathbf{]} \\ =4\cos \left(x\right){\left(2\sin \left(x\right)\cos \left(x\right)\right)}^2\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{[}\mathbf{\because }\mathbf{s}\mathbf{i}\mathbf{n}\left(2x\right)\mathbf{=}\mathbf{2}\mathbf{s}\mathbf{i}\mathbf{n}\left(x\right)\mathbf{c}\mathbf{o}\mathbf{s}\left(x\right)\mathbf{]} \\ =16{\cos }^3\left(x\right){\sin }^2\left(x\right) \end{gathered}$$

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