Differentiating $x \mapsto \cos^2(x)$ in two different ways

Question

When $\cos^2(x)$ is differentiated via the chain rule, its outcome is $-2\cos(x)\sin(x)$.

However when the double angle formula is applied to $\cos^2(x)$, it becomes $$\frac{1}{2}(1 + \cos(2x))$$ which when differentiated is just $-\sin(x)$. How can this be?

Answer 1

The derivative of the second expression is $$-\sin(2x),$$ not $-\sin(x)$ and it can because $$ \sin(2x) = 2\cos(x)\sin(x).$$

Answer 2

$$-2\sin(x)\cos(x)=-\sin(2x)$$ And your second expression's derivative is $$\frac{1}{2}(-2\sin(2x))=-\sin(2x)$$