Prove that `2 sin^2 beta + 4 cos(alpha + beta) sin alpha sin beta + cos 2(alpha + beta) = cos 2alpha`

Explanation

`2sin^(2)beta+4cos(alpha+beta)sin alpha sin beta+cos 2(alpha+beta)`.
`=2sin^(2)beta+2 cos(alpha+beta)[2sin alpha sin beta]+cos2(alpha+beta)`
`2sin^(2)beta+2cos(alpha+beta)cos(alpha-beta)-2cos^(2)(alpha+beta)`
`+2cos^(2)(alpha+beta)-1`
`=2sin^(2)beta+2(cos^(2)alpha-sin^(2)beta)-1`
`=2cos^(2)alpha-1`
`=cos 2alpha`.