Prove: (cos(x)+cos(y))^2 = 2+2cos(x+y)

I copied this up a few minutes ago incorrectly, sorry.

(cos(x) + cos(y))² ≟ 2 + 2cos(x+y)

(cos(x) + cos(y))² ≟ 2 + 2cos(x)cos(y) - 2sin(x)sin(y)

(cos(x) + cos(y))² ≟ (sin²(x) + cos²(x)) + (sin²(y) + cos²(y)) + 2cos(x)cos(y) - 2sin(x)sin(y)

(cos(x) + cos(y))² ≟ [cos²(x) + 2cos(x)cos(y) + cos²(y)] + [sin²(x) - 2sin(x)sin(y) + sin²(y)]

(cos(x) + cos(y))² ≟ [cos(x) + cos(y)]² + [sin(x) - sin(y)]²

You sure you copied it correctly this time too?

Of course they are not equal. They are equal only if sin(x) = sin(y).

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Not even close ....