Question

Упростить выражения:

1) $cos( frac{ pi }{4} +a)*cos(frac{ pi }{4}-a)+ frac{1}{2}sin^2a$

2) $cos3a+sina*sin2a$

3) $cos2a-cosa*cos3a$

Answer 1

1) $Cos( frac{ pi }{4}+ alpha )=Cos frac{ pi }{4}*Cos alpha -Sin frac{ pi }{4}*Sin alpha = frac{ sqrt{2} }{2}Cos alpha - frac{ sqrt{2}}{2}*Sin alpha$
$Cos( frac{ pi }{4}- alpha )=Cos frac{ pi }{4}*Cos alpha +Sin frac{ pi }{4}*Sin alpha = frac{ sqrt{2} }{2}Cos alpha + frac{ sqrt{2}}{2}*Sin alpha$
$(frac{ sqrt{2} }{2}Cos alpha - frac{ sqrt{2}}{2}*Sin alpha)*(frac{ sqrt{2} }{2}Cos alpha + frac{ sqrt{2}}{2}*Sin alpha)+ frac{1}{2}Sin^{2} alpha = frac{1}{2}Cos^{2} alpha$

2) Sinα*Sin2α=0,5 (Cos(-α)-Cos3α)

Cos3α+0,5Cosα-0,5Cos3α=0,5(Cosα+Cos3α)=Cosα*Cos2α

3) Cosα*Cos3α=0,5 (Cos(-2α)+Cos4α)
Cos2α-0,5Cos2α-0,5Cos4α=0,5(Cos2α-Cos4α)=Sinα*Sin3α