Need an explanation

[Loki123]

The next problem states that the three angles are inside a triangle, therefore there sum is 180 degrees. You are supposed to calculate the sum of their cos. I have a solution to this problem, but I do not understand one part. It is the part that is outlined with a red pencil. How did they turn cos alpha into that? The following writing is me trying to figure it out and failing. Any idea?

 

[blamocur]

Hint: the outlined expression is equal to [imath]\cos (180 - \phi - \gamma)[/imath]

 

[P.K.Tandon]

We know that γ = ( [ 180 - ( α + β ) ]
Hence : cos γ = cos [ 180 - ( α + β ) ] = - cos ( α + β )
Again ( α + β ) can be written as 2 ( α + β ) / 2 = [ 2 { ( α + β ) /2 } ]
Hence : cos γ = - cos [ 2 { ( α + β ) /2 } ]
Now use Identity : cos 2 (θ) = 2 cos² (θ) - 1
=> - [2 cos²( α + β ) /2 - 1 ] = - 2 cos²( α + β ) /2 +1