Question

The expression $\frac{\left(\cos 6x+6\cos 4x+15\cos 2x+10\right)}{\left(\cos 5x+5\cos 3x+10\cos x\right)}$ equal to

• A. $\cos 2x$
• B. $2\cos x$
• C. ${\cos }^{2}x$
• D. $1+\cos x$

Answer 1

The correct option is B

$2\cos x$

Simplify the given expression:

The given equation is $\frac{\left(\cos 6x+6\cos 4x+15\cos 2x+10\right)}{\left(\cos 5x+5\cos 3x+10\cos x\right)}$.

$$\begin{gathered} =\frac{\left[\cos 6x+\left(\cos 4x+5\cos 4x\right)+\left(5\cos 2x+10\cos 2x\right)+10\right]}{\left[\cos 5x+5\cos 3x+10\cos x\right]} \\ =\frac{\left[\left(\cos 6x+\cos 4x\right)+5\left(\cos 4x+\cos 2x\right)+10\left(\cos 2x+1\right)\right]}{\left(\cos 5x+5\cos 3x+10\cos x\right)} \end{gathered}$$

Using the formula $\cos A+\cos B=2\cos \left(\frac{A+B}{2}\right)\cos \left(\frac{A-B}{2}\right)$

$$\begin{gathered} \Rightarrow \frac{\left[2\cos 5x\cos x+5\left(2\cos x\cos 3x\right)+10\left(2{\cos }^{2}x\right)\right]}{\left(\cos 5x+5\cos 3x+10\cos x\right)} \\ =\frac{\left[2\cos x\left(\cos 5x+5\cos 3x+10\cos x\right)\right]}{\left(\cos 5x+5\cos 3x+10\cos x\right)} \\ =2\cos x \end{gathered}$$

Hence, the given expression can be simplified into $2\cos x$ and therefore option $\left(B\right)$ is correct.