How to prove that #2sin((4pi)/5)cos(pi/5)=sin((2pi)/5)#?

2 Answers
t0hierry
May 28, 2018

#sin((4 pi)/5) = sin (pi - pi/5) = sin(pi/5)#

Using #2sinx cosx = sin(2x)#
we obtain

#2sin(pi/5)cos(pi/5) = sin((2pi)/5)#
and finally

#2 sin((4 pi)/5)cos(pi/5) = sin((2pi)/5)#

Jim G.
May 28, 2018

#"see explanation"#

Explanation:

#"using the "color(blue)"product to sum formula"#

#•color(white)(x)2sinxcosy=sin(x+y)+sin(x-y)#

#"here "x=(4pi)/5" and "y=pi/5#

#"consider left side"#

#sin((4pi)/5+pi/5)+sin((4pi)/5-pi/5)#

#=sinpi+sin((3pi)/5)#

#=0+sin(pi-(3pi)/5)#

#=sin((2pi)/5)=" right side"rArr" verified"#

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