Prove that: `sinalpha+sinbeta+singamma-sin(alpha+beta+gamma)=4sin((alpha+beta)/2)sin((beta+gamma)/2)sin((gamma+alpha)/2)dot`

Explanation

We have to proof,

`sinalpha+sinbeta+singamma-sin(alpha+beta+gamma)=4sin((alpha+beta)/2)sin((beta+gamma)/2)sin((gamma+alpha)/2)`

Let,

L.H.S=`sinalpha+sinbeta+singamma-sin(alpha+beta+gamma)`

As we know the formula,
`sin(A^o)+sin(B^o)= 2sin(((A^o)+(B^o))/2)cos(((A^o)-(B^o))/2)` and `(sin A-sinB)= 2sin(((A^o)-(B^o))/2)cos(((A^o)+(B^o))/2)`and

`(cos A-cosB)= -2sin(((A^o)+(B^o))/2)cos(((A^o)-(B^o))/2)`
then,

L.H.S.=`2sin((alpha+beta)/2)⋅cos( (alpha−beta)/2)
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