Please solve q.26 part one? If #cot((alpha+beta)/2)+cot((beta+gamma)/2)+cot((gamma+alpha)/2)=0# **Rtp** #cosalpha+costheta+cosgamma=3cos(alpha+beta+gamma)#

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2 Answers
Mar 13, 2018

Given

#cot((alpha+beta)/2)+cot((beta+gamma)/2)+cot((gamma+alpha)/2)=0#

#=>cot((alpha+beta)/2)+cot((beta+gamma)/2)=-cot((gamma+alpha)/2)#

Multiplying both sides by #sin((alpha+beta)/2)sin((beta+gamma)/2)# we get

#=>cot((alpha+beta)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)+cot((beta+gamma)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)=-cot((gamma+alpha)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)#

#=>cos((alpha+beta)/2)sin((beta+gamma)/2)+cos((beta+gamma)/2)sin((alpha+beta)/2)=-cot((gamma+alpha)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)#

#=>sin((alpha+2beta+gamma)/2)=-(cos((gamma+alpha)/2))/sin((gamma+alpha)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)#

#=>sin((alpha+2beta+gamma)/2)*sin((gamma+alpha)/2)=-cos((gamma+alpha)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)#

Similarly we get

#sin((alpha+beta+2gamma)/2)*sin((alpha+beta)/2)=-cos((alpha+beta)/2)sin((gamma+alpha)/2)sin((beta+gamma)/2)#

So adding we get

#sin((alpha+2beta+gamma)/2)*sin((gamma+alpha)/2)+sin((alpha+beta+2gamma)/2)*sin((alpha+beta)/2)=-cos((gamma+alpha)/2)sin((alpha+beta)/2)sin((beta+gamma)/2)-cos((alpha+beta)/2)sin((gamma+alpha)/2)sin((beta+gamma)/2)#

#=>sin((alpha+2beta+gamma)/2)*sin((gamma+alpha)/2)+sin((alpha+beta+2gamma)/2)*sin((alpha+beta)/2)=-(cos((gamma+alpha)/2)sin((alpha+beta)/2)+cos((alpha+beta)/2)sin((gamma+alpha)/2))sin((beta+gamma)/2)#

#=>sin((alpha+2beta+gamma)/2)*sin((gamma+alpha)/2)+sin((alpha+beta+2gamma)/2)*sin((alpha+beta)/2)=-sin((2alpha+beta+gamma)/2)sin((beta+gamma)/2)#

#=>2sin((alpha+2beta+gamma)/2)sin((gamma+alpha)/2)+2sin((alpha+beta+2gamma)/2)sin((alpha+beta)/2)+2sin((2alpha+beta+gamma)/2)sin((beta+gamma)/2)=0#

#=>cosbeta-cos(alpha+beta+gamma)+cosgamma -cos(alpha+beta+gamma)+cos alpha -cos(alpha+beta+gamma)=0#

#=>cosalpha+costheta+cosgamma=3cos(alpha+beta+gamma)#

Proved

Mar 27, 2018

Kindly go through a Proof in the Explanation.

Explanation:

For ease of writing, let us subst.

#(alpha+beta)/2=u, (beta+gamma)/2=v, &, (gamma+alpha)/2=w#.

By Given, then, #cotu+cotv+cotw=0#.

#:. cotu+cotv=-cotw#.

#:. cosu/sinu+cosv/sinv=-cosw/sinw#.

#:. (sinvcosu+sinucosv)/(sinusinv)=-cosw/sinw, or, #

# sin(u+v)/(sinusinv)=-cosw/sinw#.

#:. sin(u+v)sinw=-sinusinvcosw, or, #

#:.2sin(u+v)sinw={-2sinusinv}cosw#.

#:. -{cos(u+v+w)-cos(u+v-w)}={cos(u+v)-cos(u-v)}cosw.#

#:. color(red)(2cos(u+v-w))color(blue)(-2cos(u+v+w))=2cos(u+v)cosw-2cos(u-v)cosw,#

#={color(blue)(cos(u+v+w))+color(red)(cos(u+v-w))}-{color(green)(cos(u-v+w))+color(magenta)(cos(u-v-w))}.#

#:. color(red)(2cos(u+v-w))-color(red)(cos(u+v-w))+color(green)(cos(u-v+w))+color(magenta)(cos(u-v-w))#
#=color(blue)(cos(u+v+w))+color(blue)(2cos(u+v+w)), i.e.,#

#color(red)(cos(u+v-w))+color(green)(cos(u-v+w))+color(magenta)(cos(u-v-w))=color(blue)(3(cos(u+v+w))#.

Here, #(u+v-w)=(alpha+beta)/2+(beta+gamma)/2- (gamma+alpha)/2=beta#,

#u-v+w=(alpha+beta)/2-(beta+gamma)/2+(gamma+alpha)/2=alpha#,

#u-v-w=(alpha+beta)/2-(beta+gamma)/2-(gamma+alpha)/2=-gamma#, and,

#u+v+w=(alpha+beta)/2+(beta+gamma)/2+(gamma+alpha)/2=alpha+beta+gamma#.

Accordingly, there follows the desired result :

#cosbeta+cosalpha+cos(-gamma)=3cos(alpha+beta+gamma), or, #

#cosbeta+cosalpha+cosgamma=3cos(alpha+beta+gamma)#.

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