Why you don't square x and y

The expression $\sin^2(x+y)$ means $(\sin(x+y))^2$. It signifies that the entire value of $\sin(x+y)$ is squared, not the individual variables $x$ or $y$ within the argument of the sine function.

Similarly, $\cos^2(x+y)$ means $(\cos(x+y))^2$.

If we were to square $x$ and $y$ individually, the expression would look like $\sin(x^2+y^2)$ or $\cos(x^2+y^2)$, which is a different mathematical operation. In this problem, the squaring applies to the result of the sine or cosine function, not to its input angles.