write the sum as a product. \n\\( \\sin ( 8 x ) + \\sin ( 6 x ) \\)

Explanation:

Step1: Recall the sum - to - product formula

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

Step2: Identify (A) and (B)

Here (A = 8x) and (B=6x).

Step3: Calculate (\frac{A + B}{2}) and (\frac{A - B}{2})

(\frac{A + B}{2}=\frac{8x+6x}{2}=\frac{14x}{2} = 7x) (\frac{A - B}{2}=\frac{8x - 6x}{2}=\frac{2x}{2}=x)

Step4: Substitute into the formula

(\sin(8x)+\sin(6x)=2\sin(7x)\cos(x))

Answer:

(2\sin(7x)\cos(x))