# How do you express cos(pi/ 3 ) * sin( ( 7 pi) / 8 )  without using products of trigonometric functions?

Use Sum or Difference
$\cos \left(\frac{\pi}{3}\right) \cdot \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \cdot \sin \left(\frac{13 \pi}{24}\right) - \frac{1}{2} \cdot \sin \left(\frac{5 \pi}{24}\right)$

#### Explanation:

This is the derivation

$\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

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

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

Now, let $x = \frac{\pi}{3}$ and $y = \frac{7 \pi}{8}$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\frac{\pi}{3} + \frac{7 \pi}{8}\right) - \sin \left(\frac{\pi}{3} - \frac{7 \pi}{8}\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\frac{8 \pi + 21 \pi}{24}\right) - \sin \left(\frac{8 \pi - 21 \pi}{24}\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\frac{29 \pi}{24}\right) - \sin \left(\frac{- 13 \pi}{24}\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\pi + \frac{5 \pi}{24}\right) - \sin \left(\frac{- 13 \pi}{24}\right)\right]$

Note: $\sin \left(\frac{- 13 \pi}{24}\right) = - \sin \left(\frac{13 \pi}{24}\right)$

so that

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\pi + \frac{5 \pi}{24}\right) - \left(- \sin \left(\frac{13 \pi}{24}\right)\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\pi + \frac{5 \pi}{24}\right) + \sin \left(\frac{13 \pi}{24}\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) =$
$\frac{1}{2} \left[\sin \pi \cos \left(\frac{5 \pi}{24}\right) + \cos \pi \sin \left(\frac{5 \pi}{24}\right) + \sin \left(\frac{13 \pi}{24}\right)\right]$

$\cos \left(\frac{\pi}{3}\right) \sin \left(\frac{7 \pi}{8}\right) = \frac{1}{2} \left[\sin \left(\frac{13 \pi}{24}\right) - \sin \left(\frac{5 \pi}{24}\right)\right]$

$\frac{1}{2} \cdot \sin \left(\frac{13 \pi}{24}\right) - \frac{1}{2} \cdot \sin \left(\frac{5 \pi}{24}\right)$