# How do you express cos( (3 pi)/ 2 ) * cos (( 3 pi) /4 )  without using products of trigonometric functions?

May 3, 2016

$\cos \left(\frac{3 \pi}{2}\right) \cdot \cos \left(\frac{3 \pi}{4}\right) = 0$

#### Explanation:

As $\cos \left(A - B\right) = \cos A \cos B - \sin A \sin B$ and $\cos \left(A + B\right) = \cos A \cos B + \sin A \sin B$, adding them

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

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

Hence $\cos \left(\frac{3 \pi}{2}\right) \cdot \cos \left(\frac{3 \pi}{4}\right)$

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

= $\frac{1}{2} \left\{\cos \left(\frac{9 \pi}{4}\right) + \cos \left(\frac{3 \pi}{4}\right)\right\}$

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

= $\frac{1}{2} \left\{\cos \left(\frac{\pi}{4}\right) - \cos \left(\frac{\pi}{4}\right)\right\} = 0$