Question

Sa se arate ca: cos$\frac{2 \pi }{7}$ + cos$\frac{4 \pi }{7}$ + cos$\frac{6 \pi }{7}$ = -$\frac{1}{2}$

Answer 1

sin$( sin\frac{ \pi }{7} *cos \frac{2 \pi }{7} + sin\frac{ \pi }{7} *cos \frac{4 \pi }{7}+ sin\frac{ \pi }{7} *cos \frac{6 \pi }{7})/sin\ \frac{ \pi }{7}$
se foloseste formula sin A+ sin B=1/2(sin(A-B)+sin(A+B))⇒
(sin(π/7-2π/7)+sin(π/7+2π/)+sin(π/7-4π/7)+sin (π/7+4π/7)+sin(π/7-6π/7)+sin(π/7+6π/7))/2sinπ/7=
(sin(-π/7)+sin(3π/7)+sin(-3π/7)+sin5π/7+sin(-5π/7)+sin 7π/7)/2sinπ/7=
(sin(-π/7)+sinπ)/2sinπ/7=(sin(-π/7)+0)/2sinπ/7=-1/2