Question

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)`

Answer 1

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

Answer 2

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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