Hello!

We need to prove this trigonometric identity. It is not hard, but we have to know some formulas. The first is the formula of cosine of the sum of two angles, which expresses `cos ( x + y )` in terms of sine and cosine of `x` and `y :`

`cos ( x + y ) = cos ( x ) cos ( y ) - sin ( x ) sin ( y ).`

The second formula does the same for sine of a sum:

`sin ( x + y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y ).`

Finally, we need the very well-known formula `cos^2 ( x ) + sin^2 ( x ) = 1.` It is actually Pythagorean theorem and of course the formula works for y, too: `cos^2 ( y ) + sin^2 ( y ) = 1.`

Now we are ready to perform the transformations:

`cos ( x + y ) cos ( y ) + sin ( x + y ) sin ( y ) =`

`= cos ( x ) cos ( y ) cos ( y ) - sin ( x ) sin ( y ) cos ( y ) + `

`+ sin ( x ) cos ( y ) sin ( y ) + cos ( x ) sin ( y ) sin ( y ).`

The second and the third summands cancel and we obtain

`cos ( x + y ) cos ( y ) + sin ( x + y ) sin ( y ) =` `= cos ( x ) cos^2 ( y ) + cos ( x ) sin^2 ( y ) =

`= cos ( x ) (cos^2 ( y ) + sin^2 ( y )) = cos(x),` which is what we want.

There is another way to prove this: use the formula for cosine of a sum in the inverse direction:

`cos ( x + y ) cos ( y ) + sin ( x + y ) sin ( y ) =`

`= cos ( x + y ) cos ( -y ) - sin ( x + y ) sin ( -y ) =`

`= cos (x + y - y) = cos(x),`

which is even more elegant.

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