Question

(a) Use the Product-to-Sum Formulas to express each product as a sum, and

(b) Use the method of adding y-coordinates to graph each function on the interval $\left[0,2\mathrm{\pi }\right]$

$G\left(x\right)=\cos \left(4x\right)\cos \left(2x\right)$

Short answer

(a) $G\left(x\right)=\frac{1}{2}\left[cos\left(6x\right)+\cos \left(2x\right)\right]$

(b)

Step-by-step solution

Step 1. Given Information

The given function is$G\left(x\right)=\cos \left(4x\right)\cos \left(2x\right)$

Part(a) Step 1. Calculation 

First multiply and divide the given function by 2 and then use the product to sum formula to obtain the required result.

$$\begin{gathered} f\left(x\right)=\frac{1}{2}\left(\cos \right(4x+2x)+\cos (4x-2x\left)\right) \\ =\frac{1}{2}\left(\cos \right(6x)+\cos (2x\left)\right) \end{gathered}$$

Part(b) Step 1. Explanation and Graph 

Graph the component functions obtained in the above part in the same coordinate system and then select several values of x at which we can compute the value of y.

Plot the points and connect them to get the required graph.