To express the difference of cosines as a product, we utilize the trigonometric identity for the difference of cosines:
[tex]\[
\cos A - \cos B = -2 \sin\left(\frac{A+B}{2}\right) \sin\left(\frac{A-B}{2}\right)
\][/tex]
Here, let [tex]\( A = 14.9b \)[/tex] and [tex]\( B = 10.3b \)[/tex].
First, calculate [tex]\( A + B \)[/tex] and [tex]\( A - B \)[/tex]:
[tex]\[
A + B = 14.9b + 10.3b = 25.2b
\][/tex]
[tex]\[
A - B = 14.9b - 10.3b = 4.6b
\][/tex]
Now, apply these results to the formula:
[tex]\[
\cos(14.9b) - \cos(10.3b) = -2 \sin\left(\frac{25.2b}{2}\right) \sin\left(\frac{4.6b}{2}\right)
\][/tex]
Simplify the arguments of the sine functions:
[tex]\[
\frac{25.2b}{2} = 12.6b
\][/tex]
[tex]\[
\frac{4.6b}{2} = 2.3b
\][/tex]
Thus, the expression becomes:
[tex]\[
\cos(14.9b) - \cos(10.3b) = -2 \sin(12.6b) \sin(2.3b)
\][/tex]
Therefore, the difference of the cosines can be written as:
[tex]\[
\cos(14.9b) - \cos(10.3b) = -2 \sin(12.6b) \sin(2.3b)
\][/tex]
