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

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Answer 1 · 2016-02-14T03:29:23

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

start with $color(blue)("Sum and Difference formulas")$

$sin (x+y)=sin x cos y + cos x sin y" " " "$ 1st equation
$sin (x-y)=sin x cos y - cos x sin y" " " "$ 2nd equation

Add 1st and 2nd equations

$sin (x+y)+sin (x-y)=2 sin x cos y$
$2sin x cos y=sin (x+y)+sin (x-y)$

$sin x cos y =1/2 sin (x+y)+1/2 sin (x-y)$

At this point let $x=pi/4$ and $y=(3pi)/4$

then use

$sin x cos y =1/2 sin (x+y)+1/2 sin (x-y)$

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

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

$sin (pi/4) cos ((3pi)/4) =1/2*( 0)+1/2 *(-1)$

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

have a nice day from the Philippines ..

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