# How do you express cos(pi/ 3 ) * sin( ( 3 pi) / 8 )  without using products of trigonometric functions?

$\cos \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{3 \pi}{8}\right) = \frac{1}{2} \cdot \sin \left(\frac{17 \pi}{24}\right) + \frac{1}{2} \cdot \sin \left(\frac{\pi}{24}\right)$

#### Explanation:

start with $\textcolor{red}{\text{Sum and Difference formulas}}$

$\sin \left(x + y\right) = \sin x \cos y + \cos x \sin y \text{ " " }$1st equation
$\sin \left(x - y\right) = \sin x \cos y - \cos x \sin y \text{ " " }$2nd equation

Subtract 2nd from the 1st equation

$\sin \left(x + y\right) - \sin \left(x - y\right) = 2 \cos x \sin y$
$2 \cos x \sin y = \sin \left(x + y\right) - \sin \left(x - y\right)$

$\cos x \sin y = \frac{1}{2} \sin \left(x + y\right) - \frac{1}{2} \sin \left(x - y\right)$

At this point let $x = \frac{\pi}{3}$ and $y = \frac{3 \pi}{8}$

then use

$\cos x \sin y = \frac{1}{2} \sin \left(x + y\right) - \frac{1}{2} \sin \left(x - y\right)$

$\cos \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{3 \pi}{8}\right) = \frac{1}{2} \cdot \sin \left(\frac{17 \pi}{24}\right) + \frac{1}{2} \cdot \sin \left(\frac{\pi}{24}\right)$

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