# How do you express cos( (4 pi)/3 ) * cos (( pi) /4 )  without using products of trigonometric functions?

I suggest , use "Sum of Trigonometric Functions"
$\cos \left(\frac{4 \pi}{3}\right) \cdot \cos \left(\frac{\pi}{4}\right) = \frac{1}{2} \cdot \cos \left(\frac{19 \pi}{12}\right) + \frac{1}{2} \cdot \cos \left(\frac{13 \pi}{12}\right)$

#### Explanation:

Use Double-Angle Formulas

$\cos \left(A + B\right) = \cos A \cos B - \sin A \sin B$

$\cos \left(A - B\right) = \cos A \cos B + \sin A \sin B$

Add the left side terms equals the sum of the right terms:

$\cos \left(A + B\right) + \cos \left(A - B\right) =$
$\cos A \cos B - \cancel{\sin A \sin B} + \cos A \cos B + \cancel{\sin A \sin B}$

$\cos \left(A + B\right) + \cos \left(A - B\right) = 2 \cdot \cos A \cos B$

it follows

$\cos A \cos B = \frac{1}{2} \cdot \cos \left(A + B\right) + \frac{1}{2} \cdot \cos \left(A - B\right)$

Use the given: Let $A = \frac{4 \pi}{3}$ and $B = \frac{\pi}{4}$

it follows

$\cos \left(\frac{4 \pi}{3}\right) \cos \left(\frac{\pi}{4}\right) =$

$\frac{1}{2} \cdot \cos \left(\frac{4 \pi}{3} + \frac{\pi}{4}\right) + \frac{1}{2} \cdot \cos \left(\frac{4 \pi}{3} - \frac{\pi}{4}\right)$

Finally, after simplification

$\cos \left(\frac{4 \pi}{3}\right) \cdot \cos \left(\frac{\pi}{4}\right) = \frac{1}{2} \cdot \cos \left(\frac{19 \pi}{12}\right) + \frac{1}{2} \cdot \cos \left(\frac{13 \pi}{12}\right)$

Have a nice day !!! from the Philippines..