# How do you express cos( (5 pi)/6 ) * cos (( pi) /6 )  without using products of trigonometric functions?

Using Sum:
$\cos \left(\frac{5 \pi}{6}\right) \cdot \cos \left(\frac{\pi}{6}\right) = \frac{1}{2} \left[\cos \left(\frac{5 \pi}{6} + \frac{\pi}{6}\right) + \cos \left(\frac{5 \pi}{6} - \frac{\pi}{6}\right)\right]$
$\cos \left(\frac{5 \pi}{6}\right) \cdot \cos \left(\frac{\pi}{6}\right) = \frac{1}{2} \left[\cos \pi + \cos \left(\frac{2 \pi}{3}\right)\right] = \frac{1}{2} \left(- 1 - \frac{1}{2}\right) = - \frac{3}{4}$

#### Explanation:

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

Add first and second equations

$\cos \left(x + y\right) + \cos \left(x - y\right) = 2 \cdot \cos x \cos y + 0$
$\cos \left(x + y\right) + \cos \left(x - y\right) = 2 \cdot \cos x \cos y$

and then

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

Let $x = \frac{5 \pi}{6}$ and $y = \frac{\pi}{6}$

$\cos \left(\frac{5 \pi}{6}\right) \cdot \cos \left(\frac{\pi}{6}\right) = \frac{1}{2} \left[\cos \left(\frac{5 \pi}{6} + \frac{\pi}{6}\right) + \cos \left(\frac{5 \pi}{6} - \frac{\pi}{6}\right)\right]$
$\cos \left(\frac{5 \pi}{6}\right) \cdot \cos \left(\frac{\pi}{6}\right) = \frac{1}{2} \left[\cos \pi + \cos \left(\frac{2 \pi}{3}\right)\right] = \frac{1}{2} \left(- 1 - \frac{1}{2}\right) = - \frac{3}{4}$

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