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

sin (pi/4) cos ((3pi)/4) =1/2 sin (pi)+1/2 sin ((-pi)/2))=-1/2

#### Explanation:

start with $\textcolor{b l u e}{\text{Sum and Difference formulas}}$

$\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

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

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

At this point let $x = \frac{\pi}{4}$ and $y = \frac{3 \pi}{4}$

then use

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

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

sin (pi/4) cos ((3pi)/4) =1/2 sin (pi)+1/2 sin ((-pi)/2))

$\sin \left(\frac{\pi}{4}\right) \cos \left(\frac{3 \pi}{4}\right) = \frac{1}{2} \cdot \left(0\right) + \frac{1}{2} \cdot \left(- 1\right)$

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

have a nice day from the Philippines ..