express the given sum as a product of sines and/or cosines. \n\\( \\cos \\frac { x } { 2 } + \\cos \\frac {…

Explanation:

Step1: Recall the sum - to - product formula

The sum - to - product formula for (\cos A+\cos B) is (2\cos\frac{A + B}{2}\cos\frac{A - B}{2}).

Step2: Identify (A) and (B)

Here (A=\frac{x}{2}) and (B = \frac{3x}{2}).

Step3: Calculate (A + B) and (A - B)

• (A + B=\frac{x}{2}+\frac{3x}{2}=\frac{x + 3x}{2}=2x).
• (A - B=\frac{x}{2}-\frac{3x}{2}=\frac{x-3x}{2}=-x).

Step4: Substitute into the formula

Since (\cos(-\theta)=\cos\theta), we have: [ \begin{align*} \cos\frac{x}{2}+\cos\frac{3x}{2}&=2\cos\frac{\frac{x}{2}+\frac{3x}{2}}{2}\cos\frac{\frac{x}{2}-\frac{3x}{2}}{2}\ &=2\cos\frac{2x}{2}\cos\frac{-x}{2}\ &=2\cos x\cos\frac{x}{2} \end{align*} ]

Answer:

(2\cos x\cos\frac{x}{2})