Prove that sin5π/18+cos4π/9=cosπ/9?

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Answer 1 · 2018-06-12T11:23:44

We know that,
#color(blue)(cosC+cosD=2cos((C+D)/2)cos((C-D)/2)to(1)#

We have to prove ,

#sin((5pi)/18)+cos((4pi)/9)=cos(pi/9)#

We take,

#LHS=sin((5pi)/18)+cos((4pi)/9)#

#color(white)(LHS)=cos(pi/2-(5pi)/18)+cos((4pi)/9)#

#color(white)(LHS)=cos((9pi-5pi)/18)+cos((4pi)/9)#

#color(white)(LHS)=color(blue)(cos((2pi)/9)+cos((4pi)/9)...toApply(1)#

#color(white)(LHS)=color(blue)(2cos(((2pi)/9+(4pi)/9)/2)cos(((2pi)/9-(4pi)/9)/2)#

#color(white)(LHS)=2cos((6pi)/18)cos((-2pi)/18)#

#color(white)(LHS)=2cos(pi/3)cos(-pi/9)#

#color(white)(LHS)=2(1/2)cos(pi/9)...to[becausecos(- theta)=costheta ]#

#color(white)(LHS)=cos(pi/9)#

#LHS=RHS#