Sin2x+sin2y=2sin(x+y)cos(x-y)

Answer 1

To prove that sin(2x) + sin(2y) = 2sin(x+y)cos(x-y), we can use the double angle formulas for sine and cosine.

Using the double angle formula for sine, sin(2x) = 2sin(x)cos(x).
Similarly, sin(2y) = 2sin(y)cos(y).

Substituting these expressions into the equation, we have:
2sin(x)cos(x) + 2sin(y)cos(y) = 2sin(x+y)cos(x-y).

Next, let's use the sum-to-product formulas for sine:
2sin(x)cos(x) = sin(2x)
2sin(y)cos(y) = sin(2y)
sin(x+y)cos(x-y) = (1/2)(sin((x+y) + (x-y))) = (1/2)(sin(2x))

Now, the equation becomes:
sin(2x) + sin(2y) = (1/2)(2sin(2x))
sin(2x) + sin(2y) = sin(2x)

This equation holds true since sin(2x) = sin(2x).

Therefore, we have proved that sin(2x) + sin(2y) = 2sin(x+y)cos(x-y).