Question

express the given sum as a product of sines and/or cosines.
\\( \cos \frac { x } { 2 } + \cos \frac { 3 x } { 2 } \\)

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:

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Answer:

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