Work on the right hand side of the equation expanding sin and cos functions. Right sides looks like this
2{(sin x cos y + cos x sin y)(cos x cos y +sin x sin y)}
Expand this using foil. Now it looks like this:
2{sinx cos x cos^2(y) + cos^2 (x)sin y cos y +sin^2 (x) sin y cos y +sin x cos x sin^2 (y) }
Factor out sin x cos x and sin y cos y from terms they appear in. Now you have
2{(sin^2(y) + cos^2 (y))sin x cos x + (sin^2(x) + cos^2 (x))sin y cos y}.
Since sin^2 (x) + cos^2 (x) = 1 the above can be simplified to:
2{sin x cos x +sin y cos y} = 2sin x cos x +2sin y cos y
But this is sin 2x + sin 2y. Proof completed :)
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