Question

Sa se arate ca dacă x+y=z atunci:

cos^2 x + cos^2 y + cos^2 z - 2cos x cos y cos z = 1

Answer 1

x+y=z Se aplica sin
$sin(x+y)=sin(z)$
$sinx*cosy+siny*cosx=sinz$ Se ridica la patrat
$sin^{2}x* cos^{2}y+ sin^{2}y* cos^{2}x+2sinx*cosy*siny*cosx= sin^{2}z$
$(1- cos^{2}x)cos^{2}y+(1- cos^{2}y)cos^{2}x+2sinx*cosy*siny*cosx=$
$1- cos^{2}z$
$cos^{2}y- cos^{2}x * cos^{2}y+ cos^{2}x- cos^{2}x * cos^{2}y$+
$2sinx*siny*cosx*cosy=1- cos^{2}z$
$cos^{2}x+ cos^{2}y+ cos^{2}z-2*cosx*cosy*(cosx*cosy-sinx*siny)$
=1
$cos^{2}x+ cos^{2}y+ cos^{2}z-2*cosx*cosy*cos(x+y)$
=1
$cos^{2}x+ cos^{2}y+ cos^{2}z-2*cosx*cosy*cosz$
=1