Question

1.Sa se arate ca $cos^{2} x+ cos^{2} 2x+ cos^{2} 3x+cos2x+cos4x+cos6x=$ 6c0sxcos2xcos3x
2.Aratati ca sinx+siny+sinz=4cosx/2*cosy/2*sinz/2 , daca x+y=z

Answer 1

Stim urmatoarele relatii
$\cos^{2}{x}=\frac{\cos{2x}+1}{2}$
$\cos{a}+\cos{b}=2\cos{\frac{a+b}{2}}\cos{\frac{a-b}{2}}$
Inlocuim cosinusurile la patrat folosind formula de mai sus
$\frac{\cos{2x}+1}{2}+\frac{\cos{4x}+1}{2}+\frac{\cos{6x}+1}{2}+\cos{2x}+\cos{4x}+\cos{6x}=\frac{3}{2}(\cos{2x}+\cos{4x}+\cos{6x})+\frac{3}{2}=\frac{3}{2}*2\cos{\frac{4x+2x}{2}}\cos{\frac{4x-2x}{2}}+\frac{3}{2}(\cos{6x}+1)=\frac{3}{2}*2\cos{3x}\cos{x}+\frac{3}{2}*2\cos^{2}{3x}=3\cos{3x}(\cos{x}+\cos{3x})=3*\cos{3x}*2\cos{\frac{3x+x}{2}}\cos{\frac{3x-x}{2}}=6\cos{3x}\cos{2x}\cos{x}$
2)
Stim formula urmatoare:
$\sin{a}+\sin{b}=2\sin{\frac{a+b}{2}}\cos{\frac{a-b}{2}}$
Tinem minte ca putem scrie pe z=x+y Atunci
Atunci avem
$\sin{x}+\sin{y}+\sin(z)=2\sin{\frac{x+y}{2}}\cos{\frac{x-y}{2}}+2\sin{\frac{z}{2}}\cos{\frac{z}{2}}=2\sin{\frac{x+y}{2}}\cos{\frac{x-y}{2}}+2\sin{\frac{x+y}{2}}\cos{\frac{x+y}{2}}=2\sin{\frac{x+y}{2}}(\cos{\frac{x-y}{2}}+\cos{\frac{x+y}{2}})$

$2*\sin{\frac{z}{2}}*2\cos{\frac{x+y+x-y}{2*2}}*\cos{\frac{x+y-(x-y)}{2*2}}=2\sin{\frac{z}{2}}*2\cos{\frac{2x}{4}}\cos{\frac{2y}{4}}=4\sin{\frac{z}{2}}\cos{\frac{x}{2}}\cos{\frac{y}{2}}$