Okay, I've been getting some of these, but I can't seem to verify this identity... any help? Here's the problem

Sin(x+y) + Sin(x-y) = 2sinxcosy

Okay, I've been working on the left side, and distribute, getting:
Sinx + Siny + Sinx - Siny

And, the sinx's add up to the 2sinx that I need for the right side, but the siny's cancel out if I don't change them around. So I changed one of them to 1/cscy... but I can't seem to work with that and the other siny to end up with cosy.

Where did I go wrong, or where do I go with it now?

Sin(x+y) = Sin(x)Cos(y) + Cos(x)Sin(y)

If you replace y by minus y and add you get twice the first term, because
cos(-y) = cos(y) and sin(-y) = - sin(y).

oh, dang, forgot about that identity!

Okay, well, adding that up it ends up as 2sinx2cosy... but I need it to be 2sinx(1)cosy.

or am I really tired and I'm thinking too out of the box to realize the entire term of 2sinxcosy consists of 2sinx's and 2cosy's?

if not, what do I switch up to drop a cosy to satisfy the right side of the equation? oh man, am i sure forgetting everything about math, lol.

To verify the given trigonometric identity:

Start with the left side of the equation: Sin(x+y) + Sin(x-y).

To simplify this expression, we can use the addition formula for sine:

Sin(a + b) = Sin(a) * Cos(b) + Cos(a) * Sin(b).
Sin(a - b) = Sin(a) * Cos(b) - Cos(a) * Sin(b).

By applying these formulas, the left side becomes:

= Sin(x) * Cos(y) + Cos(x) * Sin(y) + Sin(x) * Cos(y) - Cos(x) * Sin(y)
= 2 * Sin(x) * Cos(y)

Now, we have obtained the same expression as the right side: 2 * Sin(x) * Cos(y).

Therefore, the identity Sin(x+y) + Sin(x-y) = 2sinxcosy is verified.

In your attempt, it seems you correctly distributed the terms on the left side, resulting in Sinx + Siny + Sinx - Siny. However, instead of cancelling out the Siny terms, you should have combined them. The correct simplification is:

Sinx + Siny + Sinx - Siny = 2 * Sin(x).

Remember, when dealing with trigonometric identities, it is essential to apply the appropriate trigonometric formulas and manipulate the expressions based on those formulas.