Power Rule Isn't Sufficient

Relevant wiki: Differentiation - Chain Rule

Recall that by chain rule , we have $\dfrac{d}{dx} (f(x))^n = n \cdot f'(x) \cdot (f(x))^{n-1}$ . In this case $n=2$ and $f(x) = \sin x$ or $f(x) = \cos x$ .

And note that $\dfrac d{dx} \sin x = \cos x$ and $\dfrac d{dx} \cos x = -\sin x$ . See derivatives of trigonometric functions .

$\frac {d}{dx}\sin^{2}x=2\sin x \frac {d}{dx}\sin x=2\sin x \cos x=\sin2x$

While

$\frac {d}{dx}\cos^{2}x=2\cos x \frac {d}{dx}\cos x=2\cos x (-\sin x)=-\sin2x$

So option $2$ is right

d/dx sin^2 (x) = 2 sin (x) cos (x) = sin (2x)

d/dx sin^2 (x) = 2 sin (x) cos (x) = sin (2x)