We recognise this expression as the right hand side of:

cos(αβ) = cos α cos β + cos α cos β,

with α = x + y and β = y.

We can now write this in terms of cos(αβ) as follows:

cos(x + y)cos y + sin(x + y)sin y

= cos[(x + y) − (y)]

= cos x

We have reduced the expression to a single term.

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