Prove the identity 2(x) − 2(y) = (x+y)(x−y).

Here are the solutions.

7. Prove that $\sin^2(x) - \sin^2(y) = \sin(x+y)\sin(x-y)$.

Start with the right-hand side (RHS):

$$\sin(x+y)\sin(x-y)$$

Apply the sum and difference formulas for sine:

$$= (\sin(x)\cos(y) + \cos(x)\sin(y))(\sin(x)\cos(y) - \cos(x)\sin(y))$$

This is in the form $(a+b)(a-b) = a^2-b^2$:

$$= (\sin(x)\cos(y))^2 - (\cos(x)\sin(y))^2$$ $$= \sin^2(x)\cos^2(y) - \cos^2(x)\sin^2(y)$$

Substitute $\cos^2(y) = 1-\sin^2(y)$ and $\cos^2(x) = 1-\sin^2(x)$:

$$= \sin^2(x)(1-\sin^2(y)) - (1-\sin^2(x))\sin^2(y)$$ $$= \sin^2(x) - \sin^2(x)\sin^2(y) - \sin^2(y) + \sin^2(x)\sin^2(y)$$ $$= \sin^2(x) - \sin^2(y)$$

This matches the left-hand side (LHS). The identity is proven.

8. If $\sin(A) = 3/5$ and $\cos(B) = 5/13$, where A is in the first quadrant and B is in the fourth quadrant, find $\sin(A-B)$ and $\cos(A+B)$.